 # Solve the first order differential equation using any acceptable method. sin(x) tebollahb 2022-01-20 Answered
Solve the first order differential equation using any acceptable method.
$\mathrm{sin}\left(x\right)\frac{dy}{dx}+\left(\mathrm{cos}\left(x\right)\right)y=0$, $y\left(\frac{7\pi }{6}\right)=-2$
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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.

We have step-by-step solutions for your answer! Barbara Meeker
$\mathrm{sin}x\frac{dy}{dx}+\left(\mathrm{cos}x\right)y=0$
$y\left(\frac{7x}{6}\right)=-2$
$\frac{dy}{dx}+\left(\mathrm{cos}x\right)y=0$
$⇒\frac{dy}{dx}=-\left(\mathrm{cos}x\right)y$

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

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

We have step-by-step solutions for your answer!