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.)
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.)
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Since the root has multiplicity 2, multiply the usual undetermined-coefficients guess by \(x^2\) to clear the overlap with the homogeneous solution.
Updated On: Jul 3, 2026
- \(y_p=-ce^{3x}\)
- \(y_p=cxe^{3x}\)
- \(y_p=cx^2e^{3x}\)
- \(y_p=cx^2e^{3x}\)
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The Correct Option is C
Solution and Explanation
Step 1: Since \(3\) is a root of the characteristic equation with multiplicity \(2\), the homogeneous solution set already contains both \(e^{3x}\) and \(xe^{3x}\).
Step 2: The forcing term is \(xe^{3x}\), a degree-1 polynomial times \(e^{3x}\), so the naive undetermined-coefficients guess would be \(y_p=(Ax+B)e^{3x}\).
Step 3: This naive guess overlaps completely with the homogeneous solutions \(e^{3x}\) and \(xe^{3x}\), so it cannot work as written.
Step 4: By the modification rule, the guess must be multiplied by \(x^{s}\), where \(s\) is the multiplicity of the root, here \(s=2\). This raises the power of \(x\) attached to \(e^{3x}\) by two, so the particular solution is built on the \(x^2e^{3x}\) scale.
\[\boxed{y_p=cx^2e^{3x}}\]
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