Question

Solve the differential equation d^2 y/dx^2 = a+bx+cx^2, given that dy/dx=0 and y= d, when x=0?

Answer 1

$d^2 y/dx^2 = a+bx+cx^2$

$d y/dx = ax+\frac{1}{2}bx^2+\frac{1}{3}cx^3+C_1$

$y(x) = \frac{1}{2}ax^2+\frac{1}{6}bx^3+\frac{1}{12}cx^4+C_1x+C_2$

$dy(0)/dx=0 \Rightarrow C_1=0$

$y(0) =d \Rightarrow C_2=d$

So, solution of differential equation

$y(x) = \frac{1}{2}ax^2+\frac{1}{6}bx^3+\frac{1}{12}cx^4+d$