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.

  • $\begingroup$ no, we want a function like y = e^rx, that can be plugged into (3), which reduces (3) into its auxillary form $\endgroup$ – theenigma017 Apr 28 at 7:41
  • 1
    $\begingroup$ 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. $\endgroup$ – user88923 Apr 28 at 8:04
  • $\begingroup$ But my textbook seems too use the same basis function for non-real roots and non-distinct roots $\endgroup$ – theenigma017 Apr 28 at 9:17

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