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I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c:

$$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$

B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant

I.C $$u(x,0)=c_0(1-\cos\pi x)$$

This is new for me and I would like to know how to classify it and maybe some recommended book that includes this because mine doesn't and I think I can not separate variables. I have never seen a equation with a constant and then just one initial condition, they usually give us two.

thank you