60.2.227 problem 803
Internal
problem
ID
[10814]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
803
Date
solved
:
Tuesday, January 28, 2025 at 05:16:43 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=\frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x} \end{align*}
✓ Solution by Maple
Time used: 0.196 (sec). Leaf size: 65
dsolve(diff(y(x),x) = _F1(y(x)^2-2*ln(x))/(y(x)^2)^(1/2)/x,y(x), singsol=all)
\begin{align*}
y &= \sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_{1} \right )} \\
y &= -\sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_{1} \right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.850 (sec). Leaf size: 603
DSolve[D[y[x],x] == F1[-2*Log[x] + y[x]^2]/(x*Sqrt[y[x]^2]),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2} \text {F1}\left (K[2]^2-2 \log (x)\right )}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}+\frac {K[2]}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}-\int _1^x\left (\frac {2 K[2] \text {F1}''\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 K[2] \text {F1}''\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {4 K[2] \text {F1}''\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}''\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}''\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {2 \sqrt {K[2]^2} \text {F1}''\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}\right )dK[1]\right )dK[2]+\int _1^x\left (-\frac {\text {F1}\left (y(x)^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1]}-\frac {\sqrt {y(x)^2} \text {F1}\left (y(x)^2-2 \log (K[1])\right )}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1] y(x)}\right )dK[1]=c_1,y(x)\right ]
\]