Question

The differential equation corresponding to the general solution $y = c_{1} + (c_{2}x + c_{3})e^{2x}$ is}

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

Hint:

A constant $c$ in the solution means 0 is a root. $(Ax+B)e^{kx}$ means $k$ is a repeated root.

Answer 1

The Correct Option is C

Step 1: Concept
The roots of the auxiliary equation can be identified from the form of the general solution.

Step 2: Meaning
The term $c_{1}$ implies a root $m_{1} = 0$. The term $(c_{2}x + c_{3})e^{2x}$ implies a repeated root $m_{2}, m_{3} = 2, 2$.

Step 3: Analysis
The auxiliary equation is $m(m - 2)^{2} = 0$, which expands to $m(m^{2} - 4m + 4) = 0 \Rightarrow m^{3} - 4m^{2} + 4m = 0$.

Step 4: Conclusion
Converting the auxiliary equation back to a differential equation gives $\frac{d^{3}y}{dx^{3}} - 4\frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} = 0$. Final Answer: (C)