$(y{}^{\prime}{)}^{2}+p(x)(1+{y}^{2}{)}^{3}=0$

Aganippe76
2022-07-12
Answered

How do we solve the following differential equation?

$(y{}^{\prime}{)}^{2}+p(x)(1+{y}^{2}{)}^{3}=0$

$(y{}^{\prime}{)}^{2}+p(x)(1+{y}^{2}{)}^{3}=0$

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Hayley Mccarthy

Answered 2022-07-13
Author has **19** answers

Here is one approach.

Given:

$({y}^{\prime}{)}^{2}+p(x)({y}^{2}+1{)}^{3}=0$

We can solve for y' by taking square roots, yielding:

${y}^{\prime}=\pm \text{}i\text{}\sqrt{p(x)}\text{}({y}^{2}+1{)}^{3/2}$

We can now separate and integrate:

$\int {\displaystyle \frac{1}{({y}^{2}+1{)}^{3/2}}}\text{}dy=\pm \int i\text{}\sqrt{p(x)}\text{}dx$

Given:

$({y}^{\prime}{)}^{2}+p(x)({y}^{2}+1{)}^{3}=0$

We can solve for y' by taking square roots, yielding:

${y}^{\prime}=\pm \text{}i\text{}\sqrt{p(x)}\text{}({y}^{2}+1{)}^{3/2}$

We can now separate and integrate:

$\int {\displaystyle \frac{1}{({y}^{2}+1{)}^{3/2}}}\text{}dy=\pm \int i\text{}\sqrt{p(x)}\text{}dx$

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