Question

solve differential equation dy =2t(y2+9)dt​

Answer 1

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Answer 2

Given,

$\[dy = 2t\left( {{y^2} + 9} \right)dt\]$

Solution,

Apply variable separable method.

$\[\begin{array}{l} \Rightarrow dy = 2t\left( {{y^2} + 9} \right)dt\\ \Rightarrow \frac{{dy}}{{{y^2} + 9}} = 2tdt\end{array}\]$

Integrate the terms.

$\[\begin{array}{l} \Rightarrow \frac{1}{3}{\tan ^{ - 1}}\frac{y}{3} = 2 \times \frac{{{t^2}}}{2} + C\\ \Rightarrow \frac{1}{3}{\tan ^{ - 1}}\frac{y}{3} = {t^2} + C\\ \Rightarrow y = \frac{1}{3}\left[ {\tan \left( {3{t^2} + 3C} \right)} \right]\end{array}\]$

Hence, the solution is $\[\frac{1}{3}\left[ {\tan \left( {3{t^2} + 3C} \right)} \right].\]$