Question

In Problems$\mathbf{1}\mathbf{-}\mathbf{16}$ the indicated function${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{25}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{=}{\mathit{e}}^{\mathbf{5}\mathbf{x}}$

Short answer

The second solution of the given differential equation is$e^{-5x}$

Step-by-step solution

Definition of Reduction of order

• Reduction of order is a method for solving second-order linear ordinary differential equations in mathematics.
• When one solution is known and a second linearly independent solution is sought, this method is used.
• The approach can be used to solve n-th order equations as well.
• $y_2=y_1\left(x\right)\int \frac{e^{-\int P\left(x\right)dx}}{y_1^2\left(x\right)}dx$

Find the second solution

We have the second order differential equation

$y\text{'}\text{'}-25y=0$

as the form

$y\text{'}\text{'}+p\left(x\right)y\text{'}+q\left(x\right)y=0,\text{where}p\left(x\right)=0\text{and}q\left(x\right)=-25,$

with a first solution $y_1\left(x\right)=e^{5x}$, and we have to obtain the second solution sing the formula of reduction of order as

$$\begin{gathered} y_2\left(x\right)=y_1\left(x\right)\int \frac{e^{-\int p\left(x\right)dx}}{{\left(y_1\right)}^2}dx \\ =e^{5x}\int \frac{e^0}{{\left(e^{5x}\right)}^2}dx \\ =e^{5x}\int \frac{k}{e^{10x}}dx \\ =k{e}^{5x}\int e^{-10x}dx \\ =-\frac{k}{10}{e}^{5x}\times e^{-10x} \\ =-\frac{k}{10}{e}^{-5x} \\ =c_2{e}^{-5x} \\ =e^{-5x} \end{gathered}$$

Then, the second solution of the given differential equation $y\text{'}\text{'}-25y=0$, where we have $c_2=-\frac{k}{10}$ and we let it equal 1 .

Therefore, the second solution of the given differential equation is${\mathit{e}}^{\mathbf{-}\mathbf{5}\mathbf{x}}$