82.19.4 problem Ex. 4
Internal
problem
ID
[18851]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
IV.
Singular
solutions.
problems
at
page
43
Problem
number
:
Ex.
4
Date
solved
:
Tuesday, January 28, 2025 at 12:29:07 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+2 y^{2}+x^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.054 (sec). Leaf size: 49
dsolve(x^2*diff(y(x),x)^2-3*x*y(x)*diff(y(x),x)+2*y(x)^2+x^3=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: 60.311 (sec). Leaf size: 961
DSolve[x^2*D[y[x],x]^2-3*x*y[x]*D[y[x],x]+2*y[x]^2+x^3==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*}