29.31.7 problem 906
Internal
problem
ID
[5486]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
31
Problem
number
:
906
Date
solved
:
Monday, January 27, 2025 at 11:27:27 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+x^{3}+2 y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 49
dsolve(x^2*diff(y(x),x)^2-3*x*diff(y(x),x)*y(x)+x^3+2*y(x)^2 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -2 x^{{3}/{2}} \\
y \left (x \right ) &= 2 x^{{3}/{2}} \\
y \left (x \right ) &= \frac {x \left (c_{1}^{2}+4 x \right )}{2 c_{1}} \\
y \left (x \right ) &= \frac {x \left (c_{1}^{2} x +4\right )}{2 c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.321 (sec). Leaf size: 961
DSolve[x^2 (D[y[x],x])^2-3 x y[x] D[y[x],x]+x^3+2 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
y(x)\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
y(x)\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\
\end{align*}