Q. 50 - Calculus

Question

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

$\frac{dy}{dx}=y^3\cos x, y\left(\pi \right)=1$

Step-by-Step Solution

Answer

On solving, we get $y\left(x\right)=\frac{1}{\sqrt{1-2\sin x}}$

Step 1. Given information

Given the expression $\frac{dy}{dx}=y^3\cos x, y\left(\pi \right)=1$

Step 2: Using the variable separable method

Calculating, we get

$$\begin{gathered} \int \frac{1}{y^3}dy=\int \cos xdx \\ -\frac{1}{2y^2}=\sin x+C \\ 2y^2=-\frac{1}{\sin x+C} \end{gathered}$$

Step 3: Substitute x = π , y = 1 and solve

Calculating, we get

$$\begin{gathered} 2=-\frac{1}{0+C} \\ C=-\frac{1}{2} \\ 2y^2=-\frac{1}{\sin x-\frac{1}{2}} \\ y\left(x\right)=\frac{1}{\sqrt{1-2\sin x}} \end{gathered}$$