Q. 26 - Calculus

Question

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants. 

    

     $\frac{dy}{dx}=y\sin x$

Step-by-Step Solution

Answer

Ans:  The solution of the differential equation  $\frac{dy}{dx}=y\sin x$ is $y=A{e}^{-\cos x}$

Step 1. Given information.

given,

       $\frac{dy}{dx}=y\sin x$

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

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

Step 3. Solution

Note that the differential equation (1) is of the form of $\frac{dy}{dx}=p\left(x\right)q\left(y\right)$ in which $p\left(x\right)=\sin x$ and $q\left(y\right)=y$. So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

      $\begin{array}{r}\int \frac{1}{y}dy=\int \sin xdx\\ \ln \left|y\right|=-\cos x+C\\ y=e^{-\cos x+C}\\ =A{e}^{-\cos x}\end{array}$

Hence a solution to the differential equation $\frac{dy}{dx}=y\sin x$ is $y=A{e}^{-\cos x}$