Question

If the non-homogeneous term in an ordinary differential equation (ODE) is \(xe^{3x}\) and 3 is a root of the characteristic equation with multiplicity 2, then what is the form of the particular solution (i.e. \(y_p\))? (c is a constant.)

• A. \(y_p=-ce^{3x}\)
• B. \(y_p=cxe^{3x}\)
• C. \(y_p=cx^2e^{3x}\)
• D. \(y_p=cx^2e^{3x}\)

Hint:

Since the root has multiplicity 2, multiply the usual undetermined-coefficients guess by \(x^2\) to clear the overlap with the homogeneous solution.

Answer 1

The Correct Option is C

Step 1: List the homogeneous solutions for a double root at $3$: $y_{h,1}=e^{3x}$ and $y_{h,2}=xe^{3x}$.

Step 2: Try $k=0$: guess $y_p=(Ax+B)e^{3x}$. Both the $e^{3x}$ and $xe^{3x}$ pieces of this guess already solve the homogeneous equation, so $k=0$ fails.

Step 3: Try $k=1$: guess $y_p=x(Ax+B)e^{3x}=(Ax^2+Bx)e^{3x}$. The $Bxe^{3x}$ term still duplicates the homogeneous solution $xe^{3x}$, so $k=1$ also fails.

Step 4: Try $k=2$: guess $y_p=x^2(Ax+B)e^{3x}=(Ax^3+Bx^2)e^{3x}$. Neither $x^3e^{3x}$ nor $x^2e^{3x}$ appears in the homogeneous solution set, so $k=2$ is the smallest power that clears the duplication.

Step 5: Hence the particular solution is built on the $x^2e^{3x}$ scale, matching the option $cx^2e^{3x}$.

\[\boxed{y_p=cx^2e^{3x}}\]