60.1.486 problem 489
Internal
problem
ID
[10500]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
489
Date
solved
:
Monday, January 27, 2025 at 08:22:41 PM
CAS
classification
:
[_rational]
\begin{align*} y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a y^{2}+b x +c&=0 \end{align*}
✓ Solution by Maple
Time used: 0.253 (sec). Leaf size: 363
dsolve(y(x)^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a*y(x)^2+b*x+c = 0,y(x), singsol=all)
\begin{align*}
y &= -\frac {2 \sqrt {a \left (\left (a x -\frac {1}{2} b +x \right )^{2} a \left (a +1\right )^{2} \operatorname {RootOf}\left (-b \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a} \,a^{2}+\sqrt {-\left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right ) {\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}}}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right ) \textit {\_a}}d \textit {\_a} \right )-2 b \ln \left (2 a x -b +2 x \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\
y &= \frac {2 \sqrt {a \left (\left (a x -\frac {1}{2} b +x \right )^{2} a \left (a +1\right )^{2} \operatorname {RootOf}\left (-b \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a} \,a^{2}+\sqrt {-\left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right ) {\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}}}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right ) \textit {\_a}}d \textit {\_a} \right )-2 b \ln \left (2 a x -b +2 x \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 69.490 (sec). Leaf size: 27003
DSolve[c + b*x + a*y[x]^2 + 2*x*y[x]*D[y[x],x] + y[x]^2*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
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