Question

Solve the given initial-value problem and give

the largest interval I on which the solution is defined.

$\mathbf{sinx}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{\left(}\mathbf{cosx}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{(}\mathbf{7}\mathbf{\pi }\mathbf{/}\mathbf{6}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{2}$

Short answer

The largest interval where the solution is defined$y\left(x\right)=\frac{1}{\sin x}$

Step-by-step solution

Convert the given differential equation into variable separable form

The given differential equation can be written as

$\begin{array}{l}\sin x\frac{\mathrm{d}y}{\mathrm{d}x}=-\cos xy\\ \frac{\mathrm{d}y}{y}=-\frac{\cos x}{\sin x}\mathrm{d}x\\ \int \frac{\mathrm{d}y}{y}=-\int \frac{\cos x}{\sin x}\mathrm{d}x\end{array}$

Integrating both sides

$\begin{array}{l}\ln \left|y\right|=-\ln \left|\sin x\right|+\ln \left|c\right|\\ \ln \left|y\right|=\ln \left|\frac{c}{\sin x}\right|\\ y=\frac{c}{\sin x}\end{array}$

Find the value of c

Use the initial condition $y\left(x\right)=\frac{1}{\sin x}$to get c

$$\begin{gathered} y\left(\frac{7\pi }{6}\right)=\frac{c}{\sin \left(\frac{7\pi }{6}\right)}\left(\sin \left(\frac{7\pi }{6}\right)=-\frac{1}{2}\right) \\ -2=\frac{c}{-\frac{1}{2}} \\ c=1 \end{gathered}$$

Substitute the value c = 1 in the equation (1)

$y\left(x\right)=\frac{1}{\sin x}$