# Solve. dy/dx+3/x y = 27y^(1/3) 1n(x), x > 0

Question
Differential equations
Solve. $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+\frac{{3}}{{x}}{y}={27}{y}^{{\frac{{1}}{{3}}}}{1}{n}{\left({x}\right)},{x}{>}{0}$$

2021-02-06
Here the given differential equation is
$$\displaystyle{y}^{'}+\frac{{{3}{y}}}{{x}}={27}{y}^{{\frac{{1}}{{3}}}}$$
Let $$\displaystyle{v}={y}^{{\frac{{2}}{{3}}}}{t}{h}{e}{n}\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}=\frac{{2}}{{3}}{y}^{{\frac{{1}}{{3}}}}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$$.With respect to this transformation, the above DE reduces to $$\displaystyle\frac{{3}}{{2}}{y}^{{\frac{{1}}{{3}}}}\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+\frac{{{3}{y}}}{{x}}={27}{y}^{{\frac{{1}}{{3}}}}$$
$$\displaystyle\Rightarrow\frac{{3}}{{2}}\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+\frac{{{3}{y}^{{\frac{{2}}{{3}}}}}}{{x}}={27}$$
$$\displaystyle\Rightarrow\frac{{1}}{{2}}\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+\frac{{{y}^{{\frac{{2}}{{3}}}}}}{{x}}={9}$$
$$\displaystyle\Rightarrow\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+\frac{{{2}{y}^{{\frac{{2}}{{3}}}}}}{{x}}={18}$$
$$\displaystyle\Rightarrow\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+{2}\frac{{v}}{{x}}={18}$$
$$\displaystyle\Rightarrow{x}^{{2}}\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}+{2}{x}{v}={18}{x}^{{2}}$$
$$\displaystyle\Rightarrow\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{2}}{v}\right)}={18}{x}^{{2}}$$
$$\displaystyle\Rightarrow{x}^{{2}}{v}=\int{18}{x}^{{2}}{\left.{d}{x}\right.}+{c}$$
$$\displaystyle\Rightarrow{x}^{{2}}{v}={6}{x}^{{3}}+{c}$$
$$\displaystyle\Rightarrow{v}={6}{x}+{c}{x}^{{-{{2}}}}$$
$$\displaystyle\Rightarrow{y}^{{\frac{{2}}{{3}}}}={6}{x}+{c}{x}^{{-{{2}}}}$$
$$\displaystyle\Rightarrow{y}={\left({6}{x}+{c}{x}^{{-{{2}}}}\right)}^{{\frac{{3}}{{2}}}}$$
Where c is arbitrary constant.

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