60.2.270 problem 846
Internal
problem
ID
[10857]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
846
Date
solved
:
Tuesday, January 28, 2025 at 05:25:09 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y^{\prime }&=\frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )} \end{align*}
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 71
dsolve(diff(y(x),x) = 1/(-x+(1/y(x)+1)*x+_F1((1/y(x)+1)*x)*x^2-_F1((1/y(x)+1)*x)*x^2*(1/y(x)+1)),y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (f_{1} \left (\frac {\left (\textit {\_Z} +1\right ) x}{\textit {\_Z}}\right ) x \textit {\_Z} +f_{1} \left (\frac {\left (\textit {\_Z} +1\right ) x}{\textit {\_Z}}\right ) x -\textit {\_Z} \right ) \\
y &= {\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -\int _{}^{\frac {x \,{\mathrm e}^{\textit {\_Z}}}{{\mathrm e}^{\textit {\_Z}}-1}}\frac {1}{\textit {\_a} \left (f_{1} \left (\textit {\_a} \right ) \textit {\_a} -1\right )}d \textit {\_a} +c_{1} \right )}-1 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.558 (sec). Leaf size: 346
DSolve[D[y[x],x] == (-x + x^2*F1[x*(1 + y[x]^(-1))] + x*(1 + y[x]^(-1)) - x^2*F1[x*(1 + y[x]^(-1))]*(1 + y[x]^(-1)))^(-1),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {x \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )-1}{x \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )+x K[2] \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )-K[2]}-\int _1^x\left (\frac {\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )-\frac {K[1] \text {F1}''\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]}-\frac {K[1] \text {F1}''\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]^2}}{K[1] \left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right )-K[2]}-\frac {\left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right ) \left (K[1] \left (\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )-\frac {K[1] \text {F1}''\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]}-\frac {K[1] \text {F1}''\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]^2}\right )-1\right )}{\left (K[1] \left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right )-K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (\frac {y(x) \text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )}{K[1] \left (y(x) \text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )\right )-y(x)}-\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ]
\]