Solve D²y/dx² + 4dy/dx + 3y = 3x: A Step-by-Step Guide

Dec 10, 2025 by Alex Johnson 55 views

Understanding the Beast: What is a Second-Order Linear Differential Equation?

Have you ever encountered an equation that looks a bit intimidating, like something out of a science fiction movie, but actually holds the key to understanding how our world works? Well, that's exactly what a differential equation, especially one like d²y/dx² + 4dy/dx + 3y = 3x, represents! Don't let the symbols scare you; we're going to break it down together. At its core, a differential equation is simply an equation that relates a function with its derivatives. In our specific case, we're dealing with a second-order linear non-homogeneous differential equation. What does that mouthful mean? Let's unpack it.

Second-order means the highest derivative present is the second derivative (d²y/dx²). This often signifies systems with inertia, damping, or forces that depend on acceleration, like a bouncing spring or an electrical circuit. Understanding the order of a differential equation is the first step in choosing the right solution method, and for second-order equations, we're looking at methods involving characteristic equations and particular solutions. The presence of a second derivative means the system's behavior is influenced not just by its current state and rate of change, but also by how that rate of change is itself changing.
Linear implies that the dependent variable y and its derivatives appear only to the first power and are not multiplied together. This linearity is a huge advantage, as it allows us to use powerful, systematic methods for finding solutions, such as the principle of superposition, which lets us combine different parts of the solution. If it were non-linear, things would get much trickier and often require advanced numerical methods, making an analytical solution like the one we're pursuing far more challenging, if not impossible. The constant coefficients (1, 4, and 3) also simplify the process considerably, allowing for a straightforward characteristic equation.
Non-homogeneous refers to the 3x term on the right side of the equation. If that term were zero, it would be a homogeneous equation, which is generally simpler to solve. The 3x term represents an "external force" or input affecting our system, making it behave in a specific, forced way. This external forcing term is what differentiates the system's natural oscillations or decay from its response to an ongoing stimulus. This external input can lead to a steady-state response that persists over time, distinct from the transient behavior described by the homogeneous part.

So, when we see d²y/dx² + 4dy/dx + 3y = 3x, we're looking at a model that could describe various real-world phenomena. Imagine a spring-mass system: d²y/dx² could be acceleration, dy/dx could be velocity (representing damping), and y could be displacement. The 3x on the right might be an external force pushing or pulling the system, increasing linearly with time. Similarly, in electrical engineering, it could model an RLC circuit where current (y) or voltage (y) changes over time (x), with 3x being an external voltage source. These equations are incredibly important in physics, engineering, economics, and even biology because they describe dynamic processes – how things change over time or space. Understanding how to solve them isn't just a mathematical exercise; it's a gateway to comprehending and predicting the behavior of complex systems. The general solution to such an equation will typically be a combination of two parts: a homogeneous solution (what the system would do on its own, without external forces) and a particular solution (how the external force specifically influences the system). Getting a handle on these two components is key to unlocking the full story behind our "beast" of an equation. We'll start by tackling the easier part – the homogeneous solution – and then move on to incorporate the external influence. This systematic approach is what makes solving these equations so satisfying and useful. By breaking down the problem, we turn a seemingly complex challenge into a series of manageable steps.

The Homogeneous Solution: Cracking the "Simpler" Part

Alright, let's dive into the first crucial step to solving our differential equation, d²y/dx² + 4dy/dx + 3y = 3x: finding the homogeneous solution, often denoted as y_h. Think of y_h as the "natural" or "unforced" behavior of the system. It's what the system would do if there were no external 3x term influencing it. To find this, we temporarily set the right-hand side of our original equation to zero, effectively transforming it into its homogeneous counterpart: d²y/dx² + 4dy/dx + 3y = 0. This is where the magic of characteristic equations comes in! For linear homogeneous differential equations with constant coefficients, we can assume a solution of the form y = e^(mx), where m is a constant we need to determine. Why e^(mx)? Because the exponential function is unique in that its derivatives are simply multiples of itself, making it a perfect candidate for satisfying an equation involving a function and its derivatives. This assumption simplifies the differential equation into an algebraic one, which is much easier to solve. The general form for such an assumption works because differentiation effectively scales the exponential function, allowing us to factor it out and focus on the coefficients.

Let's plug y = e^(mx) and its derivatives into our homogeneous equation:

First derivative: dy/dx = me^(mx)
Second derivative: d²y/dx² = m²e^(mx)

Substituting these into d²y/dx² + 4dy/dx + 3y = 0 gives us: m²e^(mx) + 4me^(mx) + 3e^(mx) = 0

Since e^(mx) is never zero for any real or complex x, we can safely divide the entire equation by it, leaving us with a straightforward quadratic equation, which we call the characteristic equation:

m² + 4m + 3 = 0

Now, this is a much friendlier beast! We can solve this quadratic equation for m using various methods, such as factoring, the quadratic formula, or completing the square. Factoring is quite efficient here, as we need two numbers that multiply to 3 (the constant term) and add to 4 (the coefficient of m). Those numbers are 1 and 3.

(m + 1)(m + 3) = 0

This yields two distinct real roots for m:

m_1 = -1
m_2 = -3

These roots are critically important because they directly give us the components of our homogeneous solution. For distinct real roots, the general form of the homogeneous solution is y_h = C_1e^(m_1x) + C_2e^(m_2x). Plugging in our m values, we get:

y_h = C_1e^(-x) + C_2e^(-3x)

Here, C_1 and C_2 are arbitrary constants. They arise because our original differential equation is second-order, meaning we need two integrations to solve it, and each integration introduces a constant. These constants will ultimately be determined by initial conditions, if they were provided (e.g., y(0) and y'(0) values). This y_h describes the transient response or the natural behavior of the system. For instance, in a damped oscillation, these exponential terms would represent how the oscillations naturally decay over time without any external forcing. It's the system's inherent tendency to return to equilibrium, or to follow a specific path, if left undisturbed. These decaying exponentials suggest that without external input, the system would eventually settle down, returning to zero. Understanding how to derive this part is fundamental, as it forms the bedrock upon which we'll build our complete solution. We've successfully tamed the "simpler" part of our differential equation; now, let's move on to the more interesting challenge of the non-homogeneous term!

The Particular Solution: Handling the "Non-Homogeneous" Piece (Method of Undetermined Coefficients)

With our homogeneous solution y_h = C_1e^(-x) + C_2e^(-3x) safely tucked away, it's time to tackle the other half of our problem: finding the particular solution, denoted as y_p. This y_p accounts for the specific influence of the non-homogeneous term, the 3x on the right side of our original equation, d²y/dx² + 4dy/dx + 3y = 3x. Unlike y_h, the particular solution doesn't contain any arbitrary constants; it's a specific function that, when plugged into the differential equation, makes it true. For non-homogeneous terms that are polynomials, exponentials, sines, or cosines (or combinations thereof), the Method of Undetermined Coefficients is our go-to strategy. This method, as its name suggests, involves "guessing" the form of y_p based on g(x) (our 3x term) and then "determining" the unknown coefficients by substituting this guess back into the differential equation. The beauty of this method lies in its directness for common forcing functions, avoiding more complex integration techniques.

Since our g(x) is 3x, a first-degree polynomial, our initial guess for y_p should also be a general first-degree polynomial. This is a standard rule in the Method of Undetermined Coefficients: if g(x) is a polynomial of degree n, y_p should be a general polynomial of degree n. So, our guess is:

y_p = Ax + B

Here, A and B are the "undetermined coefficients" we need to find. Now, we need to calculate the first and second derivatives of our guessed y_p. This is essential because we'll be substituting these derivatives back into the original differential equation.

dy_p/dx = d/dx (Ax + B) = A (The derivative of Ax is A, and the derivative of a constant B is 0.)
d²y_p/dx² = d/dx (A) = 0 (The derivative of a constant A is 0.)

Next, we substitute y_p, dy_p/dx, and d²y_p/dx² back into the original non-homogeneous differential equation: d²y_p/dx² + 4dy_p/dx + 3y_p = 3x 0 + 4(A) + 3(Ax + B) = 3x

Let's simplify and rearrange the terms to group x terms and constant terms: 4A + 3Ax + 3B = 3x 3Ax + (4A + 3B) = 3x

For this equation to hold true for all values of x, the coefficients of x on both sides must be equal, and the constant terms on both sides must also be equal. This gives us a system of linear equations:

Coefficients of x: The coefficient of x on the left is 3A, and on the right is 3. Therefore: 3A = 3 From this, we easily find A = 1.
Constant terms: The constant term on the left is 4A + 3B, and on the right, there is no constant term (it's 0). Therefore: 4A + 3B = 0 Now, substitute the value of A we just found (A=1) into this equation: 4(1) + 3B = 0 4 + 3B = 0 3B = -4 B = -4/3

Fantastic! We've determined our coefficients. Now we can write down our particular solution:

y_p = (1)x + (-4/3) = x - 4/3

It's crucial to remember that sometimes, your initial guess for y_p might overlap with terms already present in your y_h. If that happens (e.g., if g(x) was e^(-x) or e^(-3x)), you would need to multiply your guess by x (or x² if needed) to ensure it's linearly independent from y_h. This is a common pitfall in the Method of Undetermined Coefficients, often called the "duplication rule" or "modification rule." However, in our case, 3x is a polynomial, and our y_h contains only exponential terms (e^(-x) and e^(-3x)), so there is no overlap, and our simple Ax + B guess worked perfectly! The particular solution describes the forced response or the steady-state behavior of the system due to the external influence. It tells us how the system specifically reacts to the 3x input. We're now just one step away from the complete picture!

Putting It All Together: The General Solution

We've done the hard work, breaking down our complex differential equation, d²y/dx² + 4dy/dx + 3y = 3x, into manageable pieces. We successfully found the homogeneous solution (y_h), which describes the system's natural behavior without external influence, and the particular solution (y_p), which accounts for the specific effect of the 3x term. Now, it's time to bring these two essential components together to form the general solution of the differential equation. The beauty of linear differential equations is that their general solution is simply the sum of the homogeneous solution and the particular solution. This is a fundamental principle known as the principle of superposition for linear equations: the total response of a system to multiple inputs (or initial conditions plus an input) is the sum of its responses to each input individually. This makes solving linear differential equations remarkably straightforward once you master finding y_h and y_p separately.

Let's recall what we found for each part:

Homogeneous Solution (y_h): We determined this by solving the characteristic equation m² + 4m + 3 = 0, yielding distinct real roots m_1 = -1 and m_2 = -3. This gave us: y_h = C_1e^(-x) + C_2e^(-3x) Remember, C_1 and C_2 are arbitrary constants. They are crucial because they allow our solution to represent an entire family of functions that satisfy the differential equation. The exact values of these constants can only be determined if we are given specific initial conditions (e.g., the value of y and dy/dx at a particular point, like x=0). Without initial conditions, C_1 and C_2 remain undetermined, hence the term "general solution." This y_h component often represents the transient behavior of a physical system, which fades away over time due to damping or resistance.
Particular Solution (y_p): We found this using the Method of Undetermined Coefficients, guessing y_p = Ax + B based on the 3x term in our original equation. After careful differentiation and substitution, we solved for A and B, and arrived at: y_p = x - 4/3 Notice that y_p has no arbitrary constants. It's a single, specific function that works for the non-homogeneous part. This y_p component typically represents the steady-state behavior or forced response of the system, which is directly sustained by the external input and does not decay over time.

Now, let's combine them to get our grand finale – the General Solution:

y(x) = C_1e^(-x) + C_2e^(-3x) + x - 4/3

This equation is the full answer to our problem! It comprehensively describes all possible functions y(x) that satisfy the original differential equation. The C_1e^(-x) + C_2e^(-3x) part often represents the transient response of a system. These exponential terms decay over time (as x increases, e^(-x) and e^(-3x) approach zero), meaning their influence diminishes. This is the system settling down. The x - 4/3 part, on the other hand, represents the steady-state response or the forced response. This is the behavior that persists indefinitely due to the continuous external forcing (3x). It describes the system's long-term behavior under the influence of the input. For example, if this represented a circuit, the e^(-x) and e^(-3x) terms would show the initial surge and decay of current when power is applied, while x - 4/3 would describe the current's behavior once the circuit has stabilized and is being driven by the external source. Understanding how these two parts interact gives us profound insight into the dynamics of the system being modeled, showing both its initial adjustments and its ultimate, forced trajectory. You've successfully navigated a fundamental concept in mathematics and engineering – congratulations!

Why This Matters: Real-World Applications and the Power of DEs

It’s truly fascinating to see how a seemingly abstract equation like d²y/dx² + 4dy/dx + 3y = 3x can unlock a deep understanding of the world around us. Differential equations aren't just theoretical constructs; they are the language of change used by scientists, engineers, economists, and even biologists to model and predict the behavior of dynamic systems. Our solved equation, y(x) = C_1e^(-x) + C_2e^(-3x) + x - 4/3, provides a perfect illustration of this power, showcasing how mathematical models can capture complex interactions and offer predictive capabilities that are invaluable in real-world scenarios.

Think about a classic RLC circuit in electrical engineering, a cornerstone of electronic design. Here, y could represent the current flowing through the circuit, and x could be time. The d²y/dx² term might relate to the inductor (representing the inertia of current, resisting changes in current flow), dy/dx to the resistor (representing damping or energy dissipation), and y to the capacitor (representing charge storage and its influence on current). The 3x term on the right could represent an externally applied voltage or current source that changes linearly with time, like a ramp function input. The exponential terms C_1e^(-x) + C_2e^(-3x) in our solution would describe the transient response – how the current behaves immediately after the circuit is switched on or off, or when a sudden change occurs. These terms show the initial fluctuations that eventually die out, thanks to the resistance (damping) in the circuit. This is the natural tendency of the circuit to settle into a stable state. The x - 4/3 part, however, represents the steady-state response or the forced response. This is the current's behavior that is sustained and driven directly by the 3x external voltage source. It tells you what the current will look like in the long run, once all the initial transients have faded away. This distinction between transient and steady-state behavior is critical for designing stable and predictable electronic devices, ensuring they perform as expected under continuous operation.

Beyond circuits, consider a damped mass-spring system, a fundamental model in mechanical engineering. Here, y might be the displacement of the mass from its equilibrium position, and x is time. The second derivative is acceleration, the first is velocity (damping by a shock absorber or air resistance), and y is displacement (spring force, Hooke's Law). The 3x term could represent an external force that increases linearly with time, perhaps a motor slowly pulling the spring or a gradually increasing wind force. Our solution would describe how the spring oscillates and then settles into a new, continuously changing position dictated by the external force. The exponential terms show the decaying oscillations, while x - 4/3 describes the underlying, steady movement caused by the external pull. Similarly, in thermal transfer, the equation could model how the temperature of an object changes over time when exposed to a linearly increasing heat source. The exponential terms would describe the initial heating or cooling dynamics, and the linear term x - 4/3 would represent the temperature profile as it approaches equilibrium with the increasing external heat. In population dynamics, similar equations can model population growth or decline influenced by external factors that change over time, such as a linearly increasing food supply. The power of solving such differential equations lies in moving beyond simply observing these phenomena to actually quantifying and predicting them. It allows engineers to design systems that are stable, physicists to understand natural laws, and economists to forecast trends. This analytical insight is what makes mastering these equations so incredibly valuable across countless scientific and practical disciplines, providing a robust framework for understanding and manipulating the dynamic aspects of our world.

Conclusion: Mastering Second-Order Linear Differential Equations

Phew! What a journey we've had, tackling the intricate world of second-order linear non-homogeneous differential equations, specifically our friend d²y/dx² + 4dy/dx + 3y = 3x. We started by understanding the fundamental components of this type of equation, recognizing its significance in modeling dynamic systems across various fields from physics to finance. We systematically broke down the solution process into two crucial, yet distinct, parts, demonstrating that even complex problems can be resolved through a structured methodology.

We mastered the art of finding the homogeneous solution (y_h) by setting the right-hand side to zero and deriving the characteristic equation. For our equation, m² + 4m + 3 = 0, we found the distinct real roots m_1 = -1 and m_2 = -3, leading to y_h = C_1e^(-x) + C_2e^(-3x). This solution captures the inherent, unforced behavior of the system, often representing transient effects that fade over time, such as the natural decay of oscillations or the initial settling of a system. The presence of arbitrary constants C_1 and C_2 reflects the family of solutions that satisfy the homogeneous part, with specific values determined by initial conditions.
Next, we conquered the challenge of the particular solution (y_p), which accounts for the specific external forcing term, 3x. Using the powerful Method of Undetermined Coefficients, we made an educated guess, y_p = Ax + B, based on the form of the forcing function. Through careful differentiation and substitution back into the original equation, we successfully determined the coefficients A=1 and B=-4/3, giving us y_p = x - 4/3. This part of the solution describes the steady-state or forced response, the long-term behavior directly driven by the input, showing how the system responds to continuous external influence without decaying.

Finally, by combining these two powerful insights, we arrived at the general solution: y(x) = C_1e^(-x) + C_2e^(-3x) + x - 4/3. This complete solution provides a comprehensive picture of all possible functions that satisfy the given differential equation, elegantly highlighting both the transient dynamics and the steady-state behavior. It's a testament to the analytical power of these mathematical tools.

Solving differential equations might seem daunting at first, but by following a structured, step-by-step approach, you can unravel even the most complex-looking expressions. The skills you've gained in understanding and solving this type of equation are incredibly valuable, forming a cornerstone for further studies in engineering, physics, mathematics, and many other scientific disciplines. Keep practicing, explore different types of forcing functions, and consider how initial conditions would refine these general solutions into specific, unique outcomes. The more you work with them, the more intuitive they become, and the more you'll appreciate their immense power in describing the changing world around us.

To deepen your understanding and explore more examples, consider visiting these trusted resources:

For a broader overview of differential equations and their applications, check out Khan Academy's Differential Equations Course.
If you're interested in the mathematical theory behind these solutions, a great resource is Paul's Online Math Notes - Differential Equations.
For a quick reference on various solution methods, Wikipedia's page on Linear Differential Equations offers a comprehensive summary.