60.1.243 problem 244
Internal
problem
ID
[10257]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
244
Date
solved
:
Monday, January 27, 2025 at 06:42:00 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x \left (2 y-x -1\right ) y^{\prime }+y \left (2 x -y-1\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 387
dsolve(x*(2*y(x)-x-1)*diff(y(x),x)+y(x)*(2*x-y(x)-1)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {3 \,5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}{40 c_{1}}+\frac {3 x 5^{{2}/{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}-x -1 \\
y &= \frac {\frac {3 \,5^{{1}/{3}} \left (-i \sqrt {3}-1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (-x -1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{1}/{3}}}{3}+x 5^{{2}/{3}} \left (i \sqrt {3}-1\right )\right )}{80}}{{\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
y &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{1}/{3}}}{3}+\left (-i \sqrt {3}-1\right ) x 5^{{2}/{3}}\right ) c_{1}}{80}}{{\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) x \,c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.752 (sec). Leaf size: 126
DSolve[x*(2*y[x]-x-1)*D[y[x],x]+y[x]*(2*x-y[x]-1)==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {\sqrt [3]{2} 5^{2/3} \left (-(K[2]-1)^3\right )^{2/3}}{27 K[2]-27 K[2]^3}dK[2]+c_1=\int _1^{\frac {(x-1) (5 x+8 y(x)+5)}{2^{2/3} \sqrt [3]{5} \sqrt [3]{-(x-1)^3} (x-2 y(x)+1)}}\frac {1}{K[1]^3+\frac {21 \sqrt [3]{-\frac {1}{2}} K[1]}{2\ 5^{2/3}}+1}dK[1],y(x)\right ]
\]