77.1.45 problem 63 (page 103)
Internal
problem
ID
[17935]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
63
(page
103)
Date
solved
:
Tuesday, January 28, 2025 at 08:28:20 PM
CAS
classification
:
[_rational]
\begin{align*} 2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 405
dsolve((2*x^3+3*x^2*y(x)+y(x)^2-y(x)^3)+( 2*y(x)^3+3*x*y(x)^2+x^2-x^3)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{2}/{3}}-12 c_{1} -12 x}{6 \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{2}/{3}}+\left (x +c_{1} \right ) \left (i \sqrt {3}-1\right )}{\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{2}/{3}}}{12}+\left (x +c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 c_{1}^{2} x +12 c_{1}^{3}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 8.468 (sec). Leaf size: 368
DSolve[(2*x^3+3*x^2*y[x]+y[x]^2-y[x]^3)+( 2*y[x]^3+3*x*y[x]^2+x^2-x^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}-\frac {\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{3 \sqrt [3]{2}} \\
y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}-6 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}} \\
y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}+6 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}} \\
y(x)\to -x \\
\end{align*}