40.2.7 problem 30
Internal
problem
ID
[6585]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
4.
Equations
of
first
order
and
first
degree
(Variable
separable).
Supplemetary
problems.
Page
22
Problem
number
:
30
Date
solved
:
Monday, January 27, 2025 at 02:14:18 PM
CAS
classification
:
[[_homogeneous, `class D`], _rational]
\begin{align*} y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (y-x y^{\prime }\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 655
dsolve(y(x)^2*(x^2+2)+(x^3+y(x)^3)*(y(x)-x*diff(y(x),x))=0,y(x), singsol=all)
\begin{align*}
y &= \frac {6 \ln \left (x \right ) x^{2}+6 c_1 \,x^{2}+\left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{2}/{3}}-6}{3 \left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{2}/{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (c_1 \,x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{2}/{3}}}{6}+\left (-i \sqrt {3}-1\right ) \left (c_1 \,x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_1^{3} x^{6}-72 c_1^{2} x^{6} \ln \left (x \right )+72 c_1^{2} x^{4}-72 c_1 \,x^{6} \ln \left (x \right )^{2}+144 c_1 \,x^{4} \ln \left (x \right )-72 c_1 \,x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 44.010 (sec). Leaf size: 423
DSolve[y[x]^2*(x^2+2)+(x^3+y[x]^3)*(y[x]-x*D[y[x],x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (2 x^2 \log (x)+(-3+2 c_1) x^2-2\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}+\frac {\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}{3 \sqrt [3]{2}} \\
y(x)\to \frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right )+2^{2/3} \left (-1+i \sqrt {3}\right ) \left (54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}\right ){}^{2/3}}{12 \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (2 x^2 \log (x)+(-3+2 c_1) x^2-2\right )}{2^{2/3} \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}{6 \sqrt [3]{2}} \\
\end{align*}