77.1.60 problem 79 (page 120)
Internal
problem
ID
[17950]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
79
(page
120)
Date
solved
:
Tuesday, January 28, 2025 at 11:16:22 AM
CAS
classification
:
[_dAlembert]
\begin{align*} y&=x {y^{\prime }}^{2}+{y^{\prime }}^{3} \end{align*}
✓ Solution by Maple
Time used: 0.132 (sec). Leaf size: 984
dsolve(y(x)=x*diff(y(x),x)^2+diff(y(x),x)^3,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\
y &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i \sqrt {3}-1\right )\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\
y &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\
\end{align*}
✓ Solution by Mathematica
Time used: 81.692 (sec). Leaf size: 1489
DSolve[y[x]==x*D[y[x],x]^2+D[y[x],x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+2 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}} \\
y(x)\to -\frac {2 x^2}{3}+\frac {1}{6} i \left (\sqrt {3}+i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+\frac {1}{96} \left (-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+c_1 \\
y(x)\to -\frac {2 x^2}{3}-\frac {1}{6} i \left (\sqrt {3}-i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}+\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+\frac {1}{96} \left (\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}-i \left (\sqrt {3}-i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+c_1 \\
y(x)\to 0 \\
\end{align*}