Question

If $y=c_1{x}^2+c_2{x}^{2}lnx,x>0,$is the general solution of a homogeneous second-order Cauchy-Euler equation, then the DE is ______.

Short answer

The differential equation is${\text{x}}^{\text{2}}\text{y'' - 3xy' + 4y = 0}$ .

Step-by-step solution

Define Cauchy-Euler equation.

A linear homogeneous ordinary differential equation with variable coefficients is known as a Euler–Cauchy equation, Cauchy–Euler equation, or simply Euler's equation in mathematics.

A solution of the differential equation is y=-5e-x+10ex.

Let the general solution be $y=c_1{x}^2+c_2{x}^{2}lnx$.

From the general solution, the obtained roots of the auxiliary solution is $m_{1,2}=2$.

Let the assumption be $y=x^m$. Differentiate with respect to $\text{x}$.

$$\begin{gathered} y\text{'}=m{x}^{(m-1)} \\ y\text{'}\text{'}=m(m-1)x^{(m-2)} \end{gathered}$$

As the roots are $m_{1,2}=2$, then the auxiliary solution is given by,