Question

the particular integral of (D²-3D+2)y=e^3x is

• A. e^3x
• B. e^2x
• C. e^x
• D. \(\frac{1}{2}\)e^3x

Answer 1

The Correct Option is A

To find the particular integral of the differential equation \((D^2 - 3D + 2)y = e^{3x}\), we'll use the method of undetermined coefficients.

1. Given Differential Equation:
\[ (D^2 - 3D + 2)y = e^{3x} \] where \( D = \frac{d}{dx} \).

2. Find the Particular Integral (P.I.):
For an exponential forcing function \( e^{ax} \), the P.I. is given by: \[ \text{P.I.} = \frac{1}{f(D)} e^{ax} = \frac{1}{f(a)} e^{ax} \quad \text{if} \quad f(a) \neq 0 \] Here, \( f(D) = D^2 - 3D + 2 \) and \( a = 3 \).

3. Evaluate \( f(3) \):
\[ f(3) = (3)^2 - 3(3) + 2 = 9 - 9 + 2 = 2 \neq 0 \]

4. Compute the Particular Integral:
Since \( f(3) = 2 \neq 0 \), we have: \[ \text{P.I.} = \frac{1}{D^2 - 3D + 2} e^{3x} = \frac{1}{2} e^{3x} \]

5. Verification:
Let's verify by substituting \( y_p = \frac{1}{2} e^{3x} \) back into the differential equation: \[ (D^2 - 3D + 2)\left(\frac{1}{2} e^{3x}\right) = \frac{1}{2} (9 - 9 + 2) e^{3x} = e^{3x} \] which matches the right-hand side of the original equation.

6. Comparing with Given Options:
The correct particular integral is: \[ e^{3x} \]

Final Answer:
The particular integral is \(\boxed{e^{3x}}\).