Solving Second-Order Linear Homogeneous ODEs with Constant Coefficients

The Core Method: From Differential to Algebraic

The type of equation we are interested in has the general form:

where $a$, $b$, and $c$ are real constants. We use $y^{\prime \prime }$ and $y^{\prime }$ as shorthand for the derivatives.

The key to solving this is to make an educated guess, or ansatz, for the solution. Since the derivatives of the exponential function $e^{rx}$ are just multiples of the function itself, it's a natural candidate. Let's try a solution of the form $y(x)=e^{rx}$. Substituting this into the equation gives:

Since $e^{rx}$ is never zero, we can divide it out, leaving us with a simple quadratic equation:

This is called the characteristic equation (or auxiliary equation). We have successfully transformed a differential equation problem into an algebraic one. The two roots of this quadratic equation, let's call them $r_1$ and $r_2$, will determine the solution to our original differential equation.

To see this method introduced, please watch the first few minutes of the following video.

The Three Cases for the Solution

The form of the general solution depends on the nature of the roots of the characteristic equation, which is determined by the discriminant $D=b^2-4ac$. This leads to three distinct cases.

The video you just started provides an excellent summary of these three cases and the form of the solution for each one.

Now, let's examine each case in detail with a worked example.

Case 1: Distinct Real Roots ($b^2-4ac>0$)

When the discriminant is positive, we get two distinct, real roots $r_1$ and $r_2$. This gives us two independent solutions, $y_1(x)=e^{r_{1}x}$ and $y_2(x)=e^{r_{2}x}$. Because our original ODE is linear and homogeneous, the principle of superposition applies: any linear combination of solutions is also a solution.

Therefore, the general solution is:

where $C_1$ and $C_2$ are arbitrary constants.

Case 2: Repeated Real Roots ($b^2-4ac=0$)

When the discriminant is zero, we get a single real root $r$ with multiplicity two. This gives us one solution, $y_1(x)=e^{rx}$. To form a general solution, we need a second, linearly independent solution. It can be shown (through a method called reduction of order) that this second solution is $y_2(x)=x{e}^{rx}$.

The general solution is then the linear combination of these two:

Case 3: Complex Conjugate Roots ($b^2-4ac<0$)

When the discriminant is negative, the roots of the characteristic equation are a complex conjugate pair, which we can write as $r=\alpha \pm i\beta$. The two solutions are $y_1(x)=e^{(\alpha +i\beta )x}$ and $y_2(x)=e^{(\alpha -i\beta )x}$.

While correct, this complex form is often inconvenient. We can find a more intuitive form using Euler's formula, which you've used before: $e^{i\theta }=\cos (\theta )+i\sin (\theta )$.

By taking linear combinations of these two complex solutions, we can form two new, real, independent solutions:

• $\frac{1}{2}(y_1+y_2)=e^{\alpha x}\cos (\beta x)$
• $\frac{1}{2i}(y_1-y_2)=e^{\alpha x}\sin (\beta x)$

The general solution is a linear combination of these real solutions:

This form is very descriptive: $\alpha$ (the real part of the root) governs the exponential growth or decay, while $\beta$ (the imaginary part) governs the frequency of oscillation. This is the mathematical origin of damped oscillations.

Finding Particular Solutions with Initial Conditions

The general solutions we've found contain two arbitrary constants, $C_1$ and $C_2$. To pin down a unique, particular solution, we need two additional pieces of information. In physics, this information often comes in the form of initial conditions, which specify the state of the system at a starting point (e.g., $x=0$). This is known as an Initial Value Problem (IVP).

Typically, you'll be given the value of the function $y(0)$ and its first derivative $y^{\prime }(0)$. The process is:

1. Find the general solution $y(x)$.
2. Find the derivative of the general solution, $y^{\prime }(x)$.
3. Plug in $x=0$ into both $y(x)$ and $y^{\prime }(x)$ and set them equal to the given initial values.
4. Solve the resulting system of two linear equations for the two constants, $C_1$ and $C_2$.

This process is demonstrated clearly in the next segment of the video.

Physical Intuition: The Damped Harmonic Oscillator

Your interest in quantum mechanics means you appreciate the physics behind the math. These ODEs aren't just abstract exercises; they are the language used to describe one of the most fundamental systems in physics: the harmonic oscillator.

Consider a mass $m$ on a spring with spring constant $k$, subject to a damping force proportional to its velocity ($-\gamma v$). Newton's second law ($F=ma$) gives:

Rearranging, we get:

This is precisely the form $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ we have been studying! The three mathematical cases correspond directly to three distinct physical behaviors.

This physical analogy is incredibly powerful. When you see a solution with real exponentials, you can think of a slow, non-oscillatory return to equilibrium. When you see a solution with sines and cosines multiplied by a decaying exponential, you can visualize an object oscillating back and forth with decreasing amplitude.

Practice Problems

To solidify your skills, let's work through a few exercises. The following resource provides a concise summary and a set of practice problems with full solutions. Try to solve them yourself before looking at the answers.

• Exercise 9: $y^{\prime \prime }+4y^{\prime }+5y=0$; with $y(0)=0$ and $y^{\prime }(0)=2$. (This is a Case 3 IVP).
• Exercise 11: $y^{\prime \prime }+6y^{\prime }+9y=0$; with $y(0)=1$ and $y^{\prime }(0)=2$. (This is a Case 2 IVP).
• Exercise 14: $\frac{d^{2}y}{d{\tau }^2}+7\frac{dy}{d\tau }+12y=0$. (This is a Case 1 general solution).

Conclusion

In this lesson, we have developed a complete, systematic method for solving any second-order linear homogeneous ordinary differential equation with constant coefficients. This powerful technique will be indispensable as we move forward.

Key Takeaways:

• We can solve the ODE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ by first solving the algebraic characteristic equation $a{r}^2+br+c=0$.
• The form of the general solution depends entirely on the nature of the roots of the characteristic equation:
  • Distinct Real Roots ($r_1,r_2$): $y(x)=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$ (Exponential behavior)
  • Repeated Real Root ($r$): $y(x)=(C_1+C_{2}x)e^{rx}$
  • Complex Roots ($\alpha \pm i\beta$): $y(x)=e^{\alpha x}(C_1\cos (\beta x)+C_2\sin (\beta x))$ (Oscillatory behavior)
• Initial conditions ($y(x_0)$ and $y^{\prime }(x_0)$) are used to determine the specific values of the constants $C_1$ and $C_2$, yielding a particular solution.
• These three mathematical cases have direct physical analogues in the motion of a damped harmonic oscillator: overdamped, critically damped, and underdamped motion.

You have now completed the mathematical preliminaries for our course. With your understanding of complex numbers, Gaussian integrals, Fourier transforms, and now second-order ODEs, you are fully equipped to engage with the core principles of quantum mechanics.

In our next lesson, we will finally introduce the master equation of non-relativistic quantum mechanics: the time-dependent Schrödinger equation. We will identify its components and begin our journey into understanding how quantum states evolve in time.