77.1.101 problem 129 (page 179)
Internal
problem
ID
[17991]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
129
(page
179)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:32 AM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} {y^{\prime \prime }}^{2}+2 x y^{\prime \prime }-y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.211 (sec). Leaf size: 1867
dsolve(diff(y(x),x$2)^2+2*x*diff(y(x),x$2)-diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (\int \frac {\left (\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}+\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right ) \left (\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}-3 \left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right )}{\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{4}+c_{2} \\
y &= \frac {\left (\int \frac {\left (\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}+\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right ) \left (\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}-3 \left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right )}{\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{4}+c_{2} \\
y &= \frac {\left (\int \frac {2 \left (\frac {\left (-1-3 i \sqrt {3}\right ) x \left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 \left (-i \sqrt {3}+\frac {1}{3}\right ) x^{2} \left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}+i \sqrt {3}\, c_{1} -i \sqrt {3}\, \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}-c_{1} +\sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right ) \left (x -\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right )}{\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= -\frac {\left (\int \frac {2 \left (-x +\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right ) \left (\frac {x \left (-1+3 i \sqrt {3}\right ) \left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}+\frac {1}{3}\right ) \left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}+i \sqrt {3}\, x^{3}-i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}-c_{1} +\sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )}{\left (x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= \frac {\left (\int \frac {2 \left (\frac {\left (-1-3 i \sqrt {3}\right ) x \left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 \left (-i \sqrt {3}+\frac {1}{3}\right ) x^{2} \left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}-i \sqrt {3}\, c_{1} -i \sqrt {3}\, \sqrt {c_{1} \left (x^{3}+c_{1} \right )}+c_{1} +\sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right ) \left (x -\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right )}{\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= -\frac {\left (\int \frac {2 \left (-x +\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right ) \left (\frac {x \left (-1+3 i \sqrt {3}\right ) \left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}+\frac {1}{3}\right ) \left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}+i \sqrt {3}\, x^{3}+i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (x^{3}+c_{1} \right )}+c_{1} +\sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )}{\left (x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= \frac {\left (\int \frac {\left (\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}+3 \left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right ) \left (\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}-\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right )}{\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{4}+c_{2} \\
y &= \frac {\left (\int \frac {\left (\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}+3 \left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right ) \left (\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}-\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}} x +x^{2}\right )}{\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{4}+c_{2} \\
y &= \frac {\left (\int \frac {2 \left (\frac {\left (-1-3 i \sqrt {3}\right ) x \left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}-\frac {1}{3}\right ) \left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}-i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (x^{3}+c_{1} \right )}+c_{1} -\sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right ) \left (x +\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right )}{\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= -\frac {\left (\int \frac {2 \left (\frac {\left (1-3 i \sqrt {3}\right ) x \left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}+\frac {1}{3}\right ) \left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}-i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (x^{3}+c_{1} \right )}-c_{1} +\sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right ) \left (x +\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right )}{\left (-x^{3}-2 c_{1} +2 \sqrt {c_{1} \left (x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= \frac {\left (\int \frac {2 \left (\frac {\left (-1-3 i \sqrt {3}\right ) x \left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}-\frac {1}{3}\right ) \left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}+i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}-c_{1} -\sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right ) \left (x +\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right )}{\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
y &= -\frac {\left (\int \frac {2 \left (x +\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}\right ) \left (\frac {\left (1-3 i \sqrt {3}\right ) x \left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}{2}+\frac {3 x^{2} \left (i \sqrt {3}+\frac {1}{3}\right ) \left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{1}/{3}}}{2}-i \sqrt {3}\, x^{3}+i \sqrt {3}\, c_{1} +i \sqrt {3}\, \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}+c_{1} +\sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )}{\left (-x^{3}+2 c_{1} +2 \sqrt {c_{1} \left (-x^{3}+c_{1} \right )}\right )^{{2}/{3}}}d x \right )}{8}+c_{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 132.325 (sec). Leaf size: 1045
DSolve[D[y[x],{x,2}]^2+2*x*D[y[x],{x,2}]-D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \int _1^x\frac {1}{4} \left (-K[1]^2+\frac {\left (K[1]^3+8 e^{3 c_1}\right ) K[1]}{\sqrt [3]{-K[1]^6+20 e^{3 c_1} K[1]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[1]^3\right ){}^3}}}+\sqrt [3]{-K[1]^6+20 e^{3 c_1} K[1]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[1]^3\right ){}^3}}\right )dK[1]+c_2 \\
y(x)\to \int _1^x\frac {1}{72} \left (-18 K[2]^2-\frac {9 i \left (-i+\sqrt {3}\right ) \left (K[2]^3+8 e^{3 c_1}\right ) K[2]}{\sqrt [3]{-K[2]^6+20 e^{3 c_1} K[2]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[2]^3\right ){}^3}}}+9 i \left (i+\sqrt {3}\right ) \sqrt [3]{-K[2]^6+20 e^{3 c_1} K[2]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[2]^3\right ){}^3}}\right )dK[2]+c_2 \\
y(x)\to \int _1^x\frac {1}{72} \left (-18 K[3]^2+\frac {9 i \left (i+\sqrt {3}\right ) \left (K[3]^3+8 e^{3 c_1}\right ) K[3]}{\sqrt [3]{-K[3]^6+20 e^{3 c_1} K[3]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[3]^3\right ){}^3}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-K[3]^6+20 e^{3 c_1} K[3]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (e^{3 c_1}-K[3]^3\right ){}^3}}\right )dK[3]+c_2 \\
y(x)\to \int _1^x\frac {1}{4} \left (-K[4]^2+\frac {\left (K[4]^3-8 e^{3 c_1}\right ) K[4]}{\sqrt [3]{-K[4]^6-20 e^{3 c_1} K[4]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[4]^3+e^{3 c_1}\right ){}^3}}}+\sqrt [3]{-K[4]^6-20 e^{3 c_1} K[4]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[4]^3+e^{3 c_1}\right ){}^3}}\right )dK[4]+c_2 \\
y(x)\to \int _1^x\frac {1}{72} \left (-18 K[5]^2+\frac {9 \left (1+i \sqrt {3}\right ) \left (8 e^{3 c_1}-K[5]^3\right ) K[5]}{\sqrt [3]{-K[5]^6-20 e^{3 c_1} K[5]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[5]^3+e^{3 c_1}\right ){}^3}}}+9 i \left (i+\sqrt {3}\right ) \sqrt [3]{-K[5]^6-20 e^{3 c_1} K[5]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[5]^3+e^{3 c_1}\right ){}^3}}\right )dK[5]+c_2 \\
y(x)\to \int _1^x\frac {1}{72} \left (-18 K[6]^2+\frac {9 i \left (i+\sqrt {3}\right ) \left (K[6]^3-8 e^{3 c_1}\right ) K[6]}{\sqrt [3]{-K[6]^6-20 e^{3 c_1} K[6]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[6]^3+e^{3 c_1}\right ){}^3}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-K[6]^6-20 e^{3 c_1} K[6]^3+8 e^{6 c_1}+8 \sqrt {e^{3 c_1} \left (K[6]^3+e^{3 c_1}\right ){}^3}}\right )dK[6]+c_2 \\
\end{align*}