60.7.45 problem 1635 (6.45)
Internal
problem
ID
[11634]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1635
(6.45)
Date
solved
:
Monday, January 27, 2025 at 11:28:40 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+a {y^{\prime }}^{2}+b y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.053 (sec). Leaf size: 79
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*y(x)=0,y(x), singsol=all)
\begin{align*}
-2 a \left (\int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 \textit {\_a} a} c_{1} a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
2 a \left (\int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 \textit {\_a} a} c_{1} a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.465 (sec). Leaf size: 332
DSolve[b*y[x] + a*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 {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2] \\
\end{align*}