Chapter 3.1, Problem 48P

Program Plan Intro

Program Description:Purpose of problem is to find differential equation if general solution of the differential equation is $y\left(x\right)=\left(c_1{e}^{x\sqrt{2}}+c_2{e}^{-x\sqrt{2}}\right)e^x$ .

Summary introduction: Problem will use the characteristic equation of homogeneous second order differential equation $a{y}^{″}+b{y}^{\prime }+cy=0$ is $a{r}^2+br+c=0$ .

If the roots are $r_1\text{ and }{r}_2$ of the characteristic equation are real and distinct, then the general solution of the differential equation is,

  $y\left(x\right)=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}$ ....... (1)

If the roots are $r_1\text{ and }{r}_2$ of the characteristic equation are real and equal, then the general solution of the differential equation is,

  $y\left(x\right)=\left(c_1+x{c}_2\right)e^{r_{1}x}$ ....... (2)