The solution of the differential equation `(d^2y)/(dx^2)=sin3x+e^x+x^2` when `y_1(0)=1` and `y(0)=0` is

Options

A. `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x-1`
B. `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x`
C. `y=(-cos3x)/3 + e^(x) + x^(4)/12+1/3x +1`
D. None of these

Answer

A

Explanation

Integrating the given differential equation, we have
`(dy)/(dx) = (-cos3x)/3+e^(x)+x^(3)/3+C_(1)`
But `y_(1)(0)=1`
So, `1=(-1/3)+C_(1)` or `C(1)=(1//3)`
Again integrating, we get
`y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x+C_(2)`
But y(0)=0, So, 0=0+1 `+C_(2)` or `C_(2)=-1`.
Thus, `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x-1`