7.5.42 problem 42
Internal
problem
ID
[146]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
42
Date
solved
:
Friday, February 07, 2025 at 07:58:30 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _exact, _rational]
\begin{align*} \frac {2 x^{{5}/{2}}-3 y^{{5}/{3}}}{2 x^{{5}/{2}} y^{{2}/{3}}}+\frac {\left (3 y^{{5}/{3}}-2 x^{{5}/{2}}\right ) y^{\prime }}{3 x^{{3}/{2}} y^{{5}/{3}}}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 181
dsolve(( (2*x^(5/2)-3*y(x)^(5/3))/( 2*x^(5/2)*y(x)^(2/3)) )+( (3*y(x)^(5/3)-2*x^(5/2))/( 3*x^(3/2)*y(x)^(5/3) ) )*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{3} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
\frac {x}{y^{{2}/{3}}}+\frac {y}{x^{{3}/{2}}}+c_1 &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 260
DSolve[( (2*x^(5/2)-3*y[x]^(5/3))/( 2*x^(5/2)*y[x]^(2/3)) )+( (3*y[x]^(5/3)-2*x^(5/2))/( 3*x^(3/2)*y[x]^(5/3) ) )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} \\
y(x)\to c_1 x^{3/2} \\
y(x)\to -\left (-\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (-\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -\left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -\sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -(-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to (-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -(-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to (-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} \\
\end{align*}