60.1.251 problem 252
Internal
problem
ID
[10265]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
252
Date
solved
:
Tuesday, January 28, 2025 at 04:26:36 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} \left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1&=0 \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 956
dsolve((x^2*y(x)-1)*diff(y(x),x)-(x*y(x)^2-1)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}}+\left (\left (-c_{1} +80\right ) x^{7}-160 x^{4}+80 x \right ) 4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}+\left (c_{1}^{2}-80 c_{1} \right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}}{x^{2} 4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}}+\left (c_{1} x^{4}-4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\
y &= \frac {\left (1-i \sqrt {3}\right ) 4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}}+x \left (2 \,4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}+x \left (1+i \sqrt {3}\right ) c_{1} \right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )}{-x^{2} \left (i \sqrt {3}-1\right ) 4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}}+\left (2 \,4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}+x^{4} \left (1+i \sqrt {3}\right ) c_{1} \right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\
y &= \frac {4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}} \left (i-\sqrt {3}\right )+x \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}+x c_{1} \left (\sqrt {3}+i\right )\right )}{x^{2} 4^{{2}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{2}/{3}} \left (i-\sqrt {3}\right )+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{{1}/{3}} {\left (c_{1} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{{1}/{3}}+x^{4} c_{1} \left (\sqrt {3}+i\right )\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 6.921 (sec). Leaf size: 127
DSolve[(x^2*y[x]-1)*D[y[x],x]-(x*y[x]^2-1)==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\frac {2 \sqrt [3]{2} 5^{2/3} \left (-\frac {1}{K[2]^3}\right )^{2/3} K[2]}{9 \left (K[2]^3-1\right )}dK[2]+c_1=\int _1^{\frac {3 x^3-4 y(x) x^2+1}{2^{2/3} \sqrt [3]{5} \sqrt [3]{-\frac {1}{x^3}} x \left (x^2 y(x)-1\right )}}\frac {1}{K[1]^3+\frac {21 \sqrt [3]{-\frac {1}{2}} K[1]}{2\ 5^{2/3}}+1}dK[1],y(x)\right ]
\]