60.1.307 problem 313
Internal
problem
ID
[10321]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
313
Date
solved
:
Sunday, March 30, 2025 at 04:09:28 PM
CAS
classification
:
[_rational]
\begin{align*} \left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.067 (sec). Leaf size: 591
ode:=(2*a*y(x)^3+3*a*x*y(x)^2-b*x^3+c*x^2)*diff(y(x),x)-a*y(x)^3+c*y(x)^2+3*b*x^2*y(x)+2*b*x^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {12^{{1}/{3}} \left (-{\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+\left (c x -c_1 \right ) a 12^{{1}/{3}}\right )}{6 {\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}} a \left (c x -c_1 \right )\right )}{12 {\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} a \left (c x -c_1 \right ) \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right )}{12 {\left (\left (-9 b \,x^{3}+9 c_1 x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 c_1 a b \,x^{4}+12 c^{3} x^{3}+81 c_1^{2} a \,x^{2}-36 c_1 \,c^{2} x^{2}+36 c_1^{2} c x -12 c_1^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
\end{align*}
✓ Mathematica. Time used: 0.338 (sec). Leaf size: 164
ode=2*b*x^3 + 3*b*x^2*y[x] + c*y[x]^2 - a*y[x]^3 + (c*x^2 - b*x^3 + 3*a*x*y[x]^2 + 2*a*y[x]^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (a x-2 a K[2]-\int _1^x\left (b+\frac {3 a K[2]^2-3 b K[2]^2-2 c K[2]}{(K[1]+K[2])^2}-\frac {2 \left (a K[2]^3-b K[2]^3-c K[2]^2\right )}{(K[1]+K[2])^3}\right )dK[1]+\frac {-a x^3+b x^3-c x^2}{(x+K[2])^2}\right )dK[2]+\int _1^x\left (-2 b K[1]+b y(x)+\frac {a y(x)^3-b y(x)^3-c y(x)^2}{(K[1]+y(x))^2}\right )dK[1]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a*y(x)**3 + 2*b*x**3 + 3*b*x**2*y(x) + c*y(x)**2 + (3*a*x*y(x)**2 + 2*a*y(x)**3 - b*x**3 + c*x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out