60.7.224 problem 1815 (book 6.224)
Internal
problem
ID
[11813]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1815
(book
6.224)
Date
solved
:
Monday, January 27, 2025 at 11:37:42 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} h \left (y\right ) y^{\prime \prime }+a h \left (y\right ) {y^{\prime }}^{2}+j \left (y\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 99
dsolve(h(y(x))*diff(diff(y(x),x),x)+a*h(y(x))*diff(y(x),x)^2+j(y(x))=0,y(x), singsol=all)
\begin{align*}
\int _{}^{y}\frac {{\mathrm e}^{2 a \textit {\_b}}}{\sqrt {{\mathrm e}^{2 a \textit {\_b}} \left (-2 \left (\int \frac {{\mathrm e}^{2 a \textit {\_b}} j \left (\textit {\_b} \right )}{h \left (\textit {\_b} \right )}d \textit {\_b} \right )+c_{1} \right )}}d \textit {\_b} -x -c_{2} &= 0 \\
-\int _{}^{y}\frac {{\mathrm e}^{2 a \textit {\_b}}}{\sqrt {{\mathrm e}^{2 a \textit {\_b}} \left (-2 \left (\int \frac {{\mathrm e}^{2 a \textit {\_b}} j \left (\textit {\_b} \right )}{h \left (\textit {\_b} \right )}d \textit {\_b} \right )+c_{1} \right )}}d \textit {\_b} -x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.569 (sec). Leaf size: 362
DSolve[j[y[x]] + a*h[y[x]]*D[y[x],x]^2 + h[y[x]]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[3]\&\right ][x+c_2] \\
\end{align*}