60.7.101 problem 1692 (book 6.101)
Internal
problem
ID
[11690]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1692
(book
6.101)
Date
solved
:
Tuesday, January 28, 2025 at 06:07:26 PM
CAS
classification
:
[NONE]
\begin{align*} \left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.095 (sec). Leaf size: 252
dsolve((a*x^2+b*x+c)^(3/2)*diff(diff(y(x),x),x)-F(y(x)/(a*x^2+b*x+c)^(1/2))=0,y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}\right ) \\
y &= \operatorname {RootOf}\left (-2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +\textit {\_g}^{2} b^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\
y &= \operatorname {RootOf}\left (2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +\textit {\_g}^{2} b^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\
\end{align*}
✓ Solution by Mathematica
Time used: 10.572 (sec). Leaf size: 251
DSolve[-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [2 a \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )+2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[3]^2+8 \int _1^{K[3]}f(K[2])dK[2]}}dK[3]&=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\
\text {Solve}\left [2 a \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )-2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[5]^2+8 \int _1^{K[5]}f(K[4])dK[4]}}dK[5]&=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\
\end{align*}