77.1.62 problem 81 (page 120)

Internal problem ID [17952]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 81 (page 120)
Date solved : Tuesday, January 28, 2025 at 11:16:25 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y&=2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \end{align*}

✓ Solution by Maple

Time used: 0.199 (sec). Leaf size: 95

dsolve(y(x)=2*diff(y(x),x)*x+y(x)^2*diff(y(x),x)^3,y(x), singsol=all)
 

\begin{align*} y &= -\frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= \frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= -\frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= \frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= 0 \\ y &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.105 (sec). Leaf size: 119

DSolve[y[x]==2*D[y[x],x]*x+y[x]^2*D[y[x],x]^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to \sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to (-1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (-1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ \end{align*}