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Q. 19

Expert-verified
Found in: Page 570

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants. $\frac{dy}{dx}={\left({x}^{3}+4\right)}^{2}$

Ans: The solution of the differential equation $\frac{dy}{dx}={\left({x}^{3}+4\right)}^{2}$ is $\frac{1}{7}{x}^{7}+2{x}^{4}+16x+C$.

See the step by step solution

## Step 1. Given information.

given,

$\frac{dy}{dx}={\left({x}^{3}+4\right)}^{2}$

## 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.

$\frac{dy}{dx}={\left({x}^{3}+4\right)}^{2}.....\left(1\right)$

## Step 3. Now,

Note that the differential equation $\left(1\right)$ does not contain the dependent variable at all, so technically the variables have already been separated. So, the differential equation can be solved by antidifferentiation. Thus, the solution of the differential equation is obtained by integrating both the sides

$\begin{array}{r}\int dy=\int {\left({x}^{3}+4\right)}^{2}dx\\ =\int \left({x}^{6}+8{x}^{3}+16\right)dx\\ =\int {x}^{6}dx+8\int {x}^{3}dx+16\int dx\\ =\frac{1}{7}{x}^{7}+2{x}^{4}+16x+C\end{array}$

Hence, a solution to the differential equation role="math" localid="1649151183048" $\frac{dy}{dx}={\left({x}^{3}+4\right)}^{2}$ is $\frac{1}{7}{x}^{7}+2{x}^{4}+16x+C\text{.}$

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