60.7.66 problem 1657 (book 6.66)
Internal
problem
ID
[11655]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1657
(book
6.66)
Date
solved
:
Tuesday, January 28, 2025 at 06:07:21 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} y^{\prime \prime }&=2 a \left (c +b x +y\right ) \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} \end{align*}
✓ Solution by Maple
Time used: 0.221 (sec). Leaf size: 795
dsolve(diff(diff(y(x),x),x)=2*a*(c+b*x+y(x))*(diff(y(x),x)^2+1)^(3/2),y(x), singsol=all)
\begin{align*}
y &= -i x +c_{1} \\
y &= i x +c_{1} \\
y &= -b x +\operatorname {RootOf}\left (c_{2} b^{3}-b^{3} x +c_{2} b -b x +\int _{}^{\textit {\_Z}}\frac {4 b^{2} a^{2} \textit {\_f}^{2} c^{2}+4 a^{2} b^{2} c \,\textit {\_f}^{3}+b^{2} a^{2} \textit {\_f}^{4}-8 b^{2} c_{1} a^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} -\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}-b^{4}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a -b^{2}}{a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1}d \textit {\_f} \right ) \\
y &= -b x +\operatorname {RootOf}\left (c_{2} b^{3}-b^{3} x +c_{2} b -b x -\int _{}^{\textit {\_Z}}-\frac {4 b^{2} a^{2} \textit {\_f}^{2} c^{2}+4 a^{2} b^{2} c \,\textit {\_f}^{3}+b^{2} a^{2} \textit {\_f}^{4}-8 b^{2} c_{1} a^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} +\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}-b^{4}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a -b^{2}}{a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 c_{1} \textit {\_f}^{2} a^{2}-8 c_{1} \textit {\_f} \,a^{2} c +4 a^{2} c_{1}^{2}-b^{2}-1}d \textit {\_f} \right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 51.157 (sec). Leaf size: 9706
DSolve[-(2*a*(c + b*x + y[x])*(1 + D[y[x],x]^2)^(3/2)) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
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