# Second order homogeneous differential equations with constant coefficients

Suppose we wish to solve the second-order homogeneous differential equation ay″ + by′ + cy = 0, (3)

where a, b, and c are constants. To solve Equation (3), we seek a function which when multiplied by a constant and added to a constant times its first derivative plus a constant times its second derivative sums identically to zero.

One function that behaves this way is the
exponential function

y = e^rx


, when r is a constant.

This is how my textbook proceeds to solve the equation (3) and it works, but is y = e^rx the only function that can solve (3) ? Then why does everybody use y=e^rx ?

For $$a=1,b=0,c=1$$, the solution of the ODE is $$C_1\cos x+ C_2 \sin x$$. So $$e^{rx}$$ is not the only kind of solution.
• No it is not, that is used to check the roots of the characteristic equation. If they are distinct and real then it would be of the form $e^{r_1x}$ and $e^{r_2x}$, otherwise you have to choose different basis functions(Like I showed in my answer). One can show that if the wronskian of the basis function is nonzero, then the solution is unique. en.wikipedia.org/wiki/Wronskian. For the case of distinct and real roots of characteristic equation with constant coefficients, the solution is always unique. – user88923 Apr 28 at 8:04