29.15.18 problem 426
Internal
problem
ID
[5024]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
15
Problem
number
:
426
Date
solved
:
Monday, January 27, 2025 at 10:04:05 AM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime } y&=\operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.572 (sec). Leaf size: 222
dsolve(y(x)*diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {4 \tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a0} \operatorname {a2} -\tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a1}^{2}-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \operatorname {a1}}{2 \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}
\]
✓ Solution by Mathematica
Time used: 0.437 (sec). Leaf size: 123
DSolve[y[x] D[y[x],x]==a0+a1 y[x]+a2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\log (\text {$\#$1} (\text {$\#$1} \text {a2}+\text {a1})+\text {a0})-\frac {2 \text {a1} \arctan \left (\frac {2 \text {$\#$1} \text {a2}+\text {a1}}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}\right )}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}}{2 \text {a2}}\&\right ][x+c_1] \\
y(x)\to \frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}}{2 \text {a2}} \\
y(x)\to -\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}}{2 \text {a2}} \\
\end{align*}