60.7.219 problem 1810 (book 6.219)
Internal
problem
ID
[11808]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1810
(book
6.219)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:27 PM
CAS
classification
:
[NONE]
\begin{align*} \left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.138 (sec). Leaf size: 336
dsolve((c+2*b*x+a*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+d*y(x)=0,y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )+a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\
y &= \operatorname {RootOf}\left (-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )-a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\
\end{align*}
✓ Solution by Mathematica
Time used: 38.051 (sec). Leaf size: 260
DSolve[d*y[x] + (c + 2*b*x + a*x^2 + y[x]^2)^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (d+\left (K[2]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\
\text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (d+\left (K[3]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\
\end{align*}