Tutorial Exercise

The indicated function $y_1(x)$ is a solution of the following differential equation,

$25y^{\text{'}\text{'}}-20y^{\text{'}}4y=0$;$,y_1=e^{2\frac{x}{5}}$

Use reduction of order, or the formula

$y_2=y_1(x){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{e^{-f(x)}dx}{y_1^2(x)}dx$

as instructed, to find a second solution $y_2(x).$

Step $1$

We are given one solution $y_1$ of the following differential equation.

$25y^{\text{'}\text{'}}-20y^{\text{'}}4y=0$;$,y_1=e^{2\frac{x}{5}}$

To find a second solution $y_2$ using the reduction of order formula

$y_2=y_1(x){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{e^{-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}P(x)dx}}{y_1^2(x)}dx$

we must write the given differential equation in standard form.

$]=[0$

SKIP $($YOU CANNOT COME BACK$)$