To solve the given second-order linear non-homogeneous differential equation:

\[
\frac{d^2 y}{dx^2} - 4y = e^{2x}
\]

We can solve this in two parts: 
1. Solve the **homogeneous equation**: 
   \[
   \frac{d^2 y}{dx^2} - 4y = 0
   \]
2. Find the **particular solution** for the non-homogeneous part: 
   \[
   \frac{d^2 y}{dx^2} - 4y = e^{2x}
   \]

### Step 1: Solve the homogeneous equation
The homogeneous equation is:
\[
\frac{d^2 y}{dx^2} - 4y = 0
\]

This can be written as a characteristic equation:
\[
r^2 - 4 = 0
\]

Solving for \(r\):
\[
r^2 = 4 \quad \Rightarrow \quad r = \pm 2
\]

So, the general solution to the homogeneous equation is:
\[
y_h(x) = C_1 e^{2x} + C_2 e^{-2x}
\]

### Step 2: Find the particular solution
To find the particular solution \(y_p(x)\), we guess a solution of the form \(y_p(x) = A x e^{2x}\) because the right-hand side is \(e^{2x}\) and \(e^{2x}\) already appears in the homogeneous solution.

Now, differentiate \(y_p(x) = A x e^{2x}\):

\[
\frac{dy_p}{dx} = A e^{2x} + 2A x e^{2x}
\]

\[
\frac{d^2 y_p}{dx^2} = 2A e^{2x} + 2A e^{2x} + 4A x e^{2x} = 4A e^{2x} + 4A x e^{2x}
\]

Substitute \(y_p(x)\) and its derivatives into the original equation:

\[
(4A e^{2x} + 4A x e^{2x}) - 4(A x e^{2x}) = e^{2x}
\]

Simplifying:
\[
4A e^{2x} = e^{2x}
\]

Equating coefficients of \(e^{2x}\):
\[
4A = 1 \quad \Rightarrow \quad A = \frac{1}{4}
\]

Thus, the particular solution is:
\[
y_p(x) = \frac{1}{4} x e^{2x}
\]

### Step 3: General solution
The general solution to the differential equation is the sum of the homogeneous and particular solutions:
\[
y(x) = C_1 e^{2x} + C_2 e^{-2x} + \frac{1}{4} x e^{2x}
\]

### Final Answer:
\[
y(x) = C_1 e^{2x} + C_2 e^{-2x} + \frac{1}{4} x e^{2x}
\]

Would you like more details on any specific step, or do you have any questions?

---

Here are five related questions:
1. How do you find the characteristic equation for higher-order linear differential equations?
2. What are the conditions when the method of undetermined coefficients is applicable?
3. How would you solve this differential equation if the non-homogeneous term was a polynomial?
4. Can the method of variation of parameters be applied to this problem?
5. What happens if the right-hand side was \(e^{3x}\) instead of \(e^{2x}\)?

**Tip:** Always verify the particular solution by substituting it back into the original equation.

Learn how to solve the second-order linear non-homogeneous differential equation d^2y/dx^2 - 4y = e^2x using the method of undetermined coefficients. This detailed solution includes steps for solving the homogeneous equation and finding the particular solution, suitable for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation