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 ${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{3}\mathit{x}$

Short answer

The second solution of the given differential equation is $\cos 3x$

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

The second order differential equation

$y\text{'}\text{'}+9y=0$

as the form

$y\text{'}\text{'}+p\left(x\right)y\text{'}+q\left(x\right)y=0,\text{then we have}p\left(x\right)=0\text{and}q\left(x\right)=9\text{,}$

with a first solution $y_1\left(x\right)=\sin 3x$, and we have to obtain the second solution using the formula of reduction of order as

$$\begin{gathered} y_2\left(x\right)=y_1\left(x\right)\int \frac{e^{-\int p\left(x\right)dx}}{{\left(y_1\right)}^2}dx \\ =\sin 3x\int \frac{e^{-\int 0dx}}{{(\sin 3x)}^2}dx \\ =\sin 3x\int \frac{k}{{\sin }^{2}3x}dx \\ =\sin 3x\int k{\csc }^{2}3xdx \\ =\sin 3x\times k\times \frac{1}{3}\cot 3x \\ =\frac{k}{3}\sin 3x\times \frac{\cos 3x}{\sin 3x} \\ =\frac{k}{3}\cos 3x \\ =c_2\cos 3x \\ =\cos 3x \end{gathered}$$

Then, the second solution of the given differential equation$y\text{'}\text{'}+9y=0$ , where $c_2=\frac{k}{3}$ and we let it equal 1 .

Therefore, the second solution of the given differential equation is $\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{3}\mathit{x}$