60.1.270 problem 271
Internal
problem
ID
[10284]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
271
Date
solved
:
Monday, January 27, 2025 at 06:50:14 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 317
dsolve((y(x)^2+x^2)*diff(y(x),x)+2*x*(y(x)+2*x)=0,y(x), singsol=all)
\begin{align*}
y &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, x^{2} \left (i \sqrt {3}-1\right )}{\left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{2}/{3}}+4 c_{1} x^{2}-\left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{2}/{3}}}{4 \left (4-16 c_{1}^{{3}/{2}} x^{3}+4 \sqrt {20 c_{1}^{3} x^{6}-8 c_{1}^{{3}/{2}} x^{3}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.216 (sec). Leaf size: 372
DSolve[(y[x]^2+x^2)*D[y[x],x]+2*x*(y[x]+2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
\end{align*}