Question

The solution of the differential equation \( (D^2 + 2)y = x^2 \) is

• A. \( C_1 \cos\sqrt{2}x + C_2 \sin\sqrt{2}x \)
• B. \( C_1 \cos\sqrt{2}x + C_2 \sin\sqrt{2}x + \frac{1}{2}(x^2 - 1) \)
• C. \( C_1 e^{\sqrt{2}x} + C_2 e^{-\sqrt{2}x} + \frac{1}{2}(x^2 - 1) \)
• D. \( C_1 e^{\sqrt{2}x} + C_2 e^{\sqrt{2}x} + (x^2 + 1) \)

Hint:

Solve LDEs by finding the complementary function and a particular integral. Match the form of the RHS.

Answer 1

The Correct Option is B

Homogeneous part: \( D^2 + 2 = 0 \Rightarrow D = \pm i\sqrt{2} \Rightarrow y_h = C_1 \cos\sqrt{2}x + C_2 \sin\sqrt{2}x \) Particular integral (RHS = \( x^2 \)): Use trial solution \( y_p = Ax^2 + Bx + C \) Substitute into LHS: \[ (D^2 + 2)y_p = 2A + 2(Ax^2 + Bx + C) = x^2 \Rightarrow 2A x^2 + 2Bx + 2C + 2A = x^2 \Rightarrow A = \frac{1}{2}, B = 0, C = -\frac{1}{2} \] So PI = \( \frac{1}{2}x^2 - \frac{1}{2} \), hence solution is: \[ y = C_1 \cos\sqrt{2}x + C_2 \sin\sqrt{2}x + \frac{1}{2}(x^2 - 1) \]