60.7.233 problem 1824 (book 6.233)
Internal
problem
ID
[11822]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1824
(book
6.233)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:29 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left ({y^{\prime }}^{2}+a \left (-y+x y^{\prime }\right )\right ) y^{\prime \prime }-b&=0 \end{align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 289
dsolve((diff(y(x),x)^2+a*(x*diff(y(x),x)-y(x)))*diff(diff(y(x),x),x)-b=0,y(x), singsol=all)
\begin{align*}
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.489 (sec). Leaf size: 281
DSolve[-b + (D[y[x],x]^2 + a*(-y[x] + x*D[y[x],x]))*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\
\text {Solve}\left [\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\
\end{align*}