Question

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29-52.

$\frac{dy}{dx}=y^2\cos x,y\left(\pi \right)=1$

Short answer

On solving, we get$y\left(x\right)=\frac{1}{1-\sin x}$

Step-by-step solution

Step 1. Given information

Given expression$\frac{dy}{dx}=y^2\cos x,y\left(\pi \right)=1$

Use the variable separable method

Calculating, we get

$$\begin{gathered} \frac{dy}{dx}=y^2\cos x \\ \frac{dy}{y^2}=\cos xdx \\ -\frac{1}{y}=\sin x+C \\ y=-\frac{1}{\sin x+C} \end{gathered}$$

Substitute x=πy=1 and calculate

Calculating, we get

$$\begin{gathered} 1=-\frac{1}{0+C} \\ C=-1 \\ y\left(x\right)=\frac{1}{1-\sin x} \end{gathered}$$