Question

The particular integral of the differential equation
\[ \frac{d^2y}{dx^2} - 6 \frac{dy}{dx} + 9y = e^{3x} \text{ is ............} \]

• A. $e^{3x}$
• B. $\dfrac{xe^{3x}}{2}$
• C. $xe^{3x}$
• D. $\dfrac{x^2 e^{3x}}{2}$

Hint:

When solving non-homogeneous differential equations, if the non-homogeneous term matches part of the solution to the homogeneous equation, multiply by a power of $x$ to get an appropriate form for the particular solution.

Answer 1

The Correct Option is D

We are given the second-order linear non-homogeneous differential equation: \[ \frac{d^2y}{dx^2} - 6 \frac{dy}{dx} + 9y = e^{3x} \] To solve this, we first solve the corresponding homogeneous equation and then find a particular solution.
1. Solve the homogeneous equation:
The homogeneous equation is: \[ \frac{d^2y}{dx^2} - 6 \frac{dy}{dx} + 9y = 0 \] The characteristic equation is: \[ r^2 - 6r + 9 = 0 \] Solving this quadratic equation: \[ (r - 3)^2 = 0 ⇒ r = 3 \] Thus, the solution to the homogeneous equation is: \[ y_h = (C_1 + C_2 x) e^{3x} \] where $C_1$ and $C_2$ are constants.
2. Find a particular solution:
Now, we find a particular solution using the method of undetermined coefficients. Since the right-hand side of the equation is $e^{3x}$, and the homogeneous solution already includes terms involving $e^{3x}$, we multiply by $x^2$ to avoid duplication in the form of the solution. We try a particular solution of the form: \[ y_p = A x^2 e^{3x} \] where $A$ is a constant to be determined.
3. Substitute $y_p$ into the original equation:
First, calculate the derivatives of $y_p$: \[ \frac{dy_p}{dx} = 2Ax e^{3x} + 3Ax^2 e^{3x} = A e^{3x} (2x + 3x^2) \] \[ \frac{d^2y_p}{dx^2} = A e^{3x} (2 + 12x + 9x^2) \] Now, substitute into the original equation: \[ \left(A e^{3x} (2 + 12x + 9x^2) \right) - 6 \left(A e^{3x} (2x + 3x^2)\right) + 9 \left(A x^2 e^{3x}\right) = e^{3x} \] Simplify the equation: \[ A e^{3x} (2 + 12x + 9x^2 - 12x - 18x^2 + 9x^2) = e^{3x} \] \[ A e^{3x} (2 + 0x + 0x^2) = e^{3x} \] Thus, we have: \[ A \times 2 = 1 ⇒ A = \frac{1}{2} \] Therefore, the particular solution is: \[ y_p = \frac{x^2 e^{3x}}{2} \] Thus, the particular integral of the given differential equation is: \[ y = y_h + y_p = (C_1 + C_2 x) e^{3x} + \frac{x^2 e^{3x}}{2} \] Therefore, the correct answer is $\dfrac{x^2 e^{3x}}{2}$.