60.7.223 problem 1814 (book 6.223)
Internal
problem
ID
[11812]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1814
(book
6.223)
Date
solved
:
Monday, January 27, 2025 at 11:37:01 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} \left (b +a \sin \left (y\right )^{2}\right ) y^{\prime \prime }+a {y^{\prime }}^{2} \cos \left (y\right ) \sin \left (y\right )+A y \left (c +a \sin \left (y\right )^{2}\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 140
dsolve((b+a*sin(y(x))^2)*diff(diff(y(x),x),x)+a*diff(y(x),x)^2*cos(y(x))*sin(y(x))+A*y(x)*(c+a*sin(y(x))^2)=0,y(x), singsol=all)
\begin{align*}
\sqrt {2}\, \left (\int _{}^{y}\frac {b +a \sin \left (\textit {\_a} \right )^{2}}{\sqrt {-\left (b +a \sin \left (\textit {\_a} \right )^{2}\right ) \left (A a \sin \left (\textit {\_a} \right )^{2}-2 A a \textit {\_a} \cos \left (\textit {\_a} \right ) \sin \left (\textit {\_a} \right )+\textit {\_a}^{2} \left (a +2 c \right ) A -2 c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-\sqrt {2}\, \left (\int _{}^{y}\frac {b +a \sin \left (\textit {\_a} \right )^{2}}{\sqrt {-\left (b +a \sin \left (\textit {\_a} \right )^{2}\right ) \left (A a \sin \left (\textit {\_a} \right )^{2}-2 A a \textit {\_a} \cos \left (\textit {\_a} \right ) \sin \left (\textit {\_a} \right )+\textit {\_a}^{2} \left (a +2 c \right ) A -2 c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.681 (sec). Leaf size: 470
DSolve[A*(c + a*Sin[y[x]]^2)*y[x] + a*Cos[y[x]]*Sin[y[x]]*D[y[x],x]^2 + (b + a*Sin[y[x]]^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {c_1+2 \int _1^{K[2]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {\cos (2 K[3]) a-a-2 b}}{\sqrt {c_1+2 \int _1^{K[3]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {2 \int _1^{K[2]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {c_1+2 \int _1^{K[2]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {\cos (2 K[3]) a-a-2 b}}{\sqrt {2 \int _1^{K[3]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {\cos (2 K[3]) a-a-2 b}}{\sqrt {c_1+2 \int _1^{K[3]}(a A K[1]+2 A c K[1]-a A \cos (2 K[1]) K[1])dK[1]}}dK[3]\&\right ][x+c_2] \\
\end{align*}