Question

Solve: y" = y'tanhx

Answer 1

set y' = z and y" = dz/dx

Thus, the differential equation becomes first order 

z' = z tanh x 

which can be solved by the method of separation of variables

dz/z = tanhxdx = (sinhx/coshx)dx

or ln|z| = ln|cosh x| + c' 

⇒ z = c₁cosh x 

or y' = c₁ cosh x 

Again, the above equation can be solved by separation of variables: 

dy = c₁ coshxdx 

⇒ y = c₁sinh x + c₂