What is the solution for y^{''}=x(y^')^2 ?

The solution for y^{''}=x(y^')^2 is y=-(2arctan(sqrt(1/(c_{1))}x))/(sqrt(c_{1))}+c_{2}

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y^{''}=x(y^')^2

y^{^{''}} = x (y^{^{'}})^{2}

y = - \frac{2 arctan ( \frac{1}{c _{1}} x )}{c _{1}} + c_{2}

Solution steps

y^{''} (x) = x (y^{'} (x))^{2}

Solve using substitution:

y = - \frac{2 arctan ( \frac{1}{c _{1}} x )}{c _{1}} + c_{2}

y = - \frac{2 arctan ( \frac{1}{c _{1}} x )}{c _{1}} + c_{2}

\frac{x}{1 + x ^{2}}, at x = 0

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\frac{2 x + 1}{x + 5} \cdot (3 x - 1)

e^{16 x}

\frac{d}{d x} (ln (x^{2} + 3))

What is the solution for y^{''}=x(y^')^2 ?
The solution for y^{''}=x(y^')^2 is y=-(2arctan(sqrt(1/(c_{1))}x))/(sqrt(c_{1))}+c_{2}