Question

Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \] is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \] find the values of $\alpha$ and $\beta$.

Hint:

If RHS term matches complementary solution, multiply trial solution by $x$. Roots of auxiliary equation directly give coefficients $\alpha$ and $\beta$.

Answer 1

Step 1: Find complementary solution roots.
From the given general solution: \[ y_c = C_1 e^{-x} + C_2 e^{2x}. \]
Thus the auxiliary equation has roots: \[ m = -1 \quad \text{and} \quad m = 2. \]
Hence the auxiliary equation is: \[ (m + 1)(m - 2) = 0. \]
\[ m^2 - m - 2 = 0. \]
Comparing with standard form: \[ m^2 + \alpha m + \beta = 0, \]
we get: \[ \alpha = -1, \] \[ \beta = -2. \]
Step 2: Verify particular solution.
Since RHS is $-e^{-x}$ and $e^{-x}$ is already part of complementary solution,
we multiply by $x$.
Hence particular solution: \[ y_p = x e^{-x}, \] which matches the given solution.
Final Answer:
\[ \boxed{\alpha = -1, \quad \beta = -2}. \]