Question

The solution y(x) of the differential equation $\frac{d^{2}y}{d{x}^2}=sin3x+e^x+x^2$ when $y_1\left(0\right)=1$ and y(0) = 0 is

• A. $-\frac{sin3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$
• B. $-\frac{sin3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x$
• C. $-\frac{cos3x}{3}+e^x+\frac{x^4}{12}+\frac{1}{3}x+1$
• D. $Noneofthese$

Answer 1

The correct option is A $-\frac{sin3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$

Integrating the given differential equation, we have $\frac{dy}{dx}=-\frac{cos3x}{3}+e^x+\frac{x^3}{3}+C_1$
but $y_1\left(0\right)$ 1 so $1=-\frac{1}{3}+1+C_1\Rightarrow C_1=\frac{1}{3}$
Again integrating, we get $y=-\frac{sin3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x+C_2$
but $y\left(0\right)=0$ so $0=1+C_2\Rightarrow C_2=-1$. Thus $y=-\frac{sin3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$