Solving $y''-k^2y=0$ without substituting $e^{kx}$ As the title says I am trying to solve $y''-k^2y=0$. The method that I want to use is to assume $y'=p$ which gives us $y''=p\frac{dp}{dy}$. 
Substituting above values in original equation gives me $p\frac{dp}{dy}-k^2y=0$ which further reduces to $\frac{dy}{dx}=\sqrt{k^2y^2+c}$. On trying to solve this differential equation I am not reaching any close to the expected answer which should be summation of two exponential terms. 

Hint: try hyperbolic substitution $$y=a\sinh u$$or$$y=a\cosh u$$ for some proper constant $a$
