Q. 49 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Given information

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

Step 2: Use the variable separable method

Calculating, we get

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

Step 3: Substitute x = π y = 1 and calculate

Calculating, we get

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