Question

Consider the following statements:
A. $(1 + e^x y + x e^x y) dx + (x e^x + 2) dy = 0$ is an exact differential equation.
B. The particular solution of $(D^2 - D - 2)y = e^{-x}$ is $-\frac{1}{3} x e^{-x}$. C. The particular solution of $(D^2 + 4)y = \sin^2 x$ is $-\frac{x}{8} \sin 2x$. D. The functions $\phi_1(x) = x^2$ and $\phi_2(x) = x |x|$ are linearly independent for $-\infty < x < \infty$.
Choose the correct answer from the options given below:

• A. A, C Only
• B. B, C Only
• C. A, B, D Only
• D. A, B, C Only

Hint:

The functions $x^2$ and $x|x|$ have Wronskian equal to $0$ everywhere, yet they are linearly independent on $\mathbb{R}$! This demonstrates that $W = 0$ does not imply linear dependence for non-analytic functions.

Answer 1

The Correct Option is C