Q. 51 - Calculus

Question

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

$\frac{dy}{dx}=\frac{x^2-2}{y^2}, y\left(0\right)=8$

Step-by-Step Solution

Answer

On solving, we get $y\left(x\right)=\sqrt[3]{x^3-6x+512}$

Step 1. Given information

Given expression $\frac{dy}{dx}=\frac{x^2-2}{y^2}, y\left(0\right)=8$

Step 2: Using the variable separable method

Calculating, we get

$$\begin{gathered} \int y^{2}dy=\int \left(x^2-2\right)dx \\ \frac{y^3}{3}=\left(\frac{1}{3}{x}^3-2x\right)+C_1 \\ {y}^3=x^3-6x+C \\ y=\sqrt[3]{x^3-6x+C} \end{gathered}$$

Step 3: Substitute x = 0 y = 8 and calculate

Calculating, we get

$$\begin{gathered} 8=\sqrt[3]{C} \\ C=512 \\ y\left(x\right)=\sqrt[3]{x^3-6x+512} \end{gathered}$$