Double Integrate to Solve t+(d2t/dr2) = -5sin(r)

[Mathpower]

Please help me double integrate this... (so that I will have t = ... )
t+(d2t/dr2) = -5sin(r)

Thank you...
Also any advice appreciated as to how to go about solving 'separate the variable type questions'.

 

[HallsofIvy]

This is
$$\frac{d^2r}{dt^2}= -t- 5sin(r)$$?

First, it is not a matter of directly integrating nor is this a "separable" equation. Rather, this is a "linear second order differential equation with constant coefficients". The corresponding homogeneous equation is $d^2r/dt^2+ t= 0$. It's characteristic equation is $\lambda^2+ 1= 0$ which has roots $\pm i$. That means the general solution to the corresponding homogeneous equation is of the for $r= C_1cos(t)+ C_2sin(t)$. To complete the solution, start with $r= At cos(t)+ Bt sin(t)$. Differentiate that twice, put it into the equation, and find A and B so that it satifies the equation. The add that to the previous general equation.

(I have the feeling, from your post, that you might not recognise any of those words. If you have not studied "linear differential equations", where did you get this problem?)