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Q. 19

Expert-verifiedFound in: Page 570

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 $\frac{1}{7}{x}^{7}+2{x}^{4}+16x+C$.

given,

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

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

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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