29.32.24 problem 958
Internal
problem
ID
[5536]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
958
Date
solved
:
Monday, January 27, 2025 at 11:35:50 AM
CAS
classification
:
[_quadrature]
\begin{align*} \left (1-a y\right ) {y^{\prime }}^{2}&=a y \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 399
dsolve((1-a*y(x))*diff(y(x),x)^2 = a*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \frac {\operatorname {RootOf}\left (4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \,\operatorname {csgn}\left (a \right ) a c_{1} \operatorname {RootOf}\left (-4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} +4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )+4 \,\operatorname {csgn}\left (a \right ) a x \operatorname {RootOf}\left (-4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} +4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )+\operatorname {RootOf}\left (-4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} +4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a} \\
y \left (x \right ) &= \frac {\operatorname {RootOf}\left (4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \,\operatorname {csgn}\left (a \right ) a c_{1} \operatorname {RootOf}\left (4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} -4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )-4 \,\operatorname {csgn}\left (a \right ) a x \operatorname {RootOf}\left (4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} -4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )+\operatorname {RootOf}\left (4 \,\operatorname {csgn}\left (a \right ) a c_{1} \textit {\_Z} -4 \,\operatorname {csgn}\left (a \right ) a x \textit {\_Z} +4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+\textit {\_Z}^{2}-\cos \left (\textit {\_Z} \right )^{2}\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.911 (sec). Leaf size: 115
DSolve[(1-a y[x]) (D[y[x],x])^2==a y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\arcsin \left (\sqrt {\text {$\#$1}} \sqrt {a}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [-\sqrt {a} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\arcsin \left (\sqrt {\text {$\#$1}} \sqrt {a}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [\sqrt {a} x+c_1\right ] \\
y(x)\to 0 \\
\end{align*}