60.7.20 problem 1610 (6.20)
Internal
problem
ID
[11609]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1610
(6.20)
Date
solved
:
Tuesday, January 28, 2025 at 06:06:59 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }&=\frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}} \end{align*}
✓ Solution by Maple
Time used: 0.065 (sec). Leaf size: 92
dsolve(diff(diff(y(x),x),x)=x^(-3/2)*f(y(x)*x^(-1/2)),y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (\textit {\_Z} \,x^{{3}/{2}}+4 f \left (\frac {\textit {\_Z}}{\sqrt {x}}\right ) x^{2}\right ) \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2} \right ) \sqrt {x} \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2} \right ) \sqrt {x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.158 (sec). Leaf size: 754
DSolve[-(f[y[x]*x^(-1/2)]*x^(-3/2)) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\int _1^{y(x)}\frac {2}{\sqrt {x} \sqrt {\frac {K[3]^2+4 x c_1+8 x \int _1^{\frac {K[3]}{\sqrt {x}}}f(K[2])dK[2]}{x}}}dK[3]-\int _1^x\left (\frac {2 \left (\frac {y(x)}{2 \sqrt {K[4]}}-\frac {\sqrt {\frac {y(x)^2}{2 K[4]}+2 c_1+4 \int _1^{\frac {y(x)}{\sqrt {K[4]}}}f(K[2])dK[2]}}{\sqrt {2}}\right )}{K[4] \sqrt {\frac {y(x)^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {y(x)}{\sqrt {K[4]}}}f(K[2])dK[2]}{K[4]}}}+\int _1^{y(x)}\left (-\frac {\frac {4 c_1+8 \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]-\frac {4 f\left (\frac {K[3]}{\sqrt {K[4]}}\right ) K[3]}{\sqrt {K[4]}}}{K[4]}-\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]}{K[4]^2}}{\sqrt {K[4]} \left (\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]}{K[4]}\right ){}^{3/2}}-\frac {1}{K[4]^{3/2} \sqrt {\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]}{K[4]}}}\right )dK[3]\right )dK[4]&=c_2,y(x)\right ] \\
\text {Solve}\left [\int _1^{y(x)}-\frac {2}{\sqrt {x} \sqrt {\frac {K[5]^2+4 x c_1+8 x \int _1^{\frac {K[5]}{\sqrt {x}}}f(K[2])dK[2]}{x}}}dK[5]-\int _1^x\left (\int _1^{y(x)}\left (\frac {\frac {4 c_1+8 \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]-\frac {4 f\left (\frac {K[5]}{\sqrt {K[6]}}\right ) K[5]}{\sqrt {K[6]}}}{K[6]}-\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]}{K[6]^2}}{\sqrt {K[6]} \left (\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]}{K[6]}\right ){}^{3/2}}+\frac {1}{K[6]^{3/2} \sqrt {\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]}{K[6]}}}\right )dK[5]-\frac {2 \left (\frac {y(x)}{2 \sqrt {K[6]}}+\frac {\sqrt {\frac {y(x)^2}{2 K[6]}+2 c_1+4 \int _1^{\frac {y(x)}{\sqrt {K[6]}}}f(K[2])dK[2]}}{\sqrt {2}}\right )}{K[6] \sqrt {\frac {y(x)^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {y(x)}{\sqrt {K[6]}}}f(K[2])dK[2]}{K[6]}}}\right )dK[6]&=c_2,y(x)\right ] \\
\end{align*}