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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
State True or False for the following:
The solution of $\frac{\text{dy}}{\text{dx}}=\Big(\frac{\text{y}}{\text{x}}\Big)^{\frac{1}{3}}$ is $\text{y}^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}=\text{C}.$
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Answer

True.Solution:
We have $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^{\frac{1}{3}}}{\text{x}^{\frac{1}{3}}}$
$\Rightarrow\text{y}^{-\frac{1}{3}}\text{dy}=\text{x}^{-\frac{1}{3}}\text{dx}$
$\Rightarrow\int\text{y}^{-\frac{1}{3}}\text{dy}=\int\text{x}^{-\frac{1}{3}}\text{dx}$
$\Rightarrow\frac{3}{2}\text{y}^{\frac{2}{3}}=\frac{3}{2}\text{x}^{\frac{2}{3}}+\text{C}'$
$\Rightarrow\text{y}^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}=\text{C}$

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