60.7.50 problem 1640 (6.50)
Internal
problem
ID
[11639]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1640
(6.50)
Date
solved
:
Monday, January 27, 2025 at 11:28:54 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }+a y {y^{\prime }}^{2}+b y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 70
dsolve(diff(diff(y(x),x),x)+a*y(x)*diff(y(x),x)^2+b*y(x)=0,y(x), singsol=all)
\begin{align*}
a \left (\int _{}^{y}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-a \left (\int _{}^{y}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.110 (sec). Leaf size: 290
DSolve[b*y[x] + a*y[x]*D[y[x],x]^2 + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\
\end{align*}