Question

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

$\frac{dy}{dx}=y\sin x$

Short answer

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

Step-by-step solution

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

role="math" localid="1649178088966" $\begin{array}{r}\int \frac{1}{y}dy=\int \sin xdx\\ \ln \left|y\right|=-\cos x+C\\ y=e^{-\cos x+C}\\ =A{e}^{-\cos x}\end{array}$

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