# Solve the first order differential equation using any acceptable method. sin(x)

Solve the first order differential equation using any acceptable method.
$\mathrm{sin}\left(x\right)\frac{dy}{dx}+\left(\mathrm{cos}\left(x\right)\right)y=0$, $y\left(\frac{7\pi }{6}\right)=-2$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

otoplilp1
We have given First Order Differential Equation,

Now, We are using Variable Separable Method for solving the First Order Linear Differential Equation is as follows:
First, we need to separate the variable with the respective derivatives:

$\frac{dy}{y}=-\frac{\mathrm{cos}x}{\mathrm{sin}x}dx$
Now, Integrating both sides of the above equation:
$\int \frac{dy}{y}=-\int \frac{\mathrm{cos}x}{\mathrm{sin}x}dx$
$\mathrm{ln}|y|=-\mathrm{ln}|\mathrm{sin}x|+\mathrm{ln}|C|$
$\mathrm{ln}y=\mathrm{ln}\left(\frac{C}{\mathrm{sin}x}\right)$
$y\mathrm{sin}x=C$
Now, We are applying the given Initial Condition is as follow:
$y\left(\frac{7\pi }{6}\right)=-2$
$-2×\mathrm{sin}\left(\frac{7\pi }{6}\right)=C$

$C=2$
The resulting curve of the First Order Linear Differential Equation is y sinx =2.

Barbara Meeker
$\mathrm{sin}x\frac{dy}{dx}+\left(\mathrm{cos}x\right)y=0$
$y\left(\frac{7x}{6}\right)=-2$
$\frac{dy}{dx}+\left(\mathrm{cos}x\right)y=0$
$⇒\frac{dy}{dx}=-\left(\mathrm{cos}x\right)y$

$⇒\mathrm{ln}y=-\mathrm{ln}\left(\mathrm{sin}x\right)+\mathrm{ln}C$
$⇒\mathrm{ln}y+\mathrm{ln}\mathrm{sin}x=\mathrm{ln}C$
$⇒y\left(\mathrm{sin}x\right)=C$
$y\left(\frac{7x}{6}\right)=-2$
$-2\mathrm{sin}\left(\frac{7x}{6}\right)=C$

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