Question

Which of the following differential equation is satisfied by $y_{1}(x)=e^{x}$, $y_{2}(x)=x~e^{x}$ and $y_{3}=e^{2x}?$

• A. $\frac{d^{3}y}{dx^{3}}+\frac{4~d^{2}y}{dx^{2}}+\frac{5~dy}{dx}+2y=0$
• B. $\frac{d^{3}y}{dx^{3}}-\frac{4d^{2}y}{dx^{2}}+\frac{5dy}{dx}-2y=0$
• C. $\frac{d^{3}y}{dx^{3}}+\frac{4~d^{2}y}{dx^{2}}-\frac{5dy}{dx}-2y=0$
• D. $\frac{d^{3}y}{dx^{3}}-\frac{4~d^{2}y}{dx^{2}}+\frac{5~dy}{dx}+2y=0$

Hint:

If you see a term $x^k e^{ax}$ in a solution, it always means the root $a$ is repeated at least $k+1$ times. Reconstructing the polynomial from roots is the most reliable way to find the original differential equation.

Answer 1

The Correct Option is B