60.1.393 problem 404
Internal
problem
ID
[10407]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
404
Date
solved
:
Sunday, March 30, 2025 at 04:37:13 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} a {y^{\prime }}^{2}+b \,x^{2} y^{\prime }+c x y&=0 \end{align*}
✓ Maple. Time used: 0.190 (sec). Leaf size: 481
ode:=a*diff(y(x),x)^2+b*x^2*diff(y(x),x)+c*x*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
-2 a \int _{}^{y}\frac {1+3 \left (b \,x^{3}+6 \textit {\_f} a -\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x \right ) \int _{\textit {\_b}}^{x}\frac {-\textit {\_a}^{4} b^{2}+2 \textit {\_a} \textit {\_f} a c +\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a}^{2} b}{\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \left (b \,\textit {\_a}^{3}-\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} +6 \textit {\_f} a \right )^{2}}d \textit {\_a}}{b \,x^{3}+6 \textit {\_f} a -\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x}d \textit {\_f} -\int _{\textit {\_b}}^{x}\frac {-b \,\textit {\_a}^{2}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y a c \right )}}{-b \,\textit {\_a}^{3}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y a c \right )}\, \textit {\_a} -6 y a}d \textit {\_a} +c_1 &= 0 \\
-2 a \int _{}^{y}\frac {1+3 \left (b \,x^{3}+\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x +6 \textit {\_f} a \right ) \int _{\textit {\_b}}^{x}\frac {\textit {\_a} \left (\textit {\_a}^{3} b^{2}+\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} b -2 \textit {\_f} a c \right )}{\left (b \,\textit {\_a}^{3}+\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} +6 \textit {\_f} a \right )^{2} \sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}}d \textit {\_a}}{b \,x^{3}+\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x +6 \textit {\_f} a}d \textit {\_f} -\int _{\textit {\_b}}^{x}\frac {b \,\textit {\_a}^{2}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y a c \right )}}{b \,\textit {\_a}^{3}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y a c \right )}\, \textit {\_a} +6 y a}d \textit {\_a} +c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 6.18 (sec). Leaf size: 313
ode=c*x*y[x] + b*x^2*D[y[x],x] + a*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {-6 b \text {arctanh}\left (\frac {b x \sqrt {b^2 x^4-4 a c x y(x)}}{b^2 x^3-4 a c y(x)}\right )+(6 b+4 c) \text {arctanh}\left (\frac {x^2 (3 b+2 c)}{3 \sqrt {b^2 x^4-4 a c x y(x)}}\right )+(3 b+2 c) \log \left (9 a y(x)+3 b x^3+c x^3\right )}{6 (3 b+c)}+\frac {b \log (6 b y(x)+2 c y(x))}{2 (3 b+c)}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {6 b \text {arctanh}\left (\frac {b x \sqrt {b^2 x^4-4 a c x y(x)}}{b^2 x^3-4 a c y(x)}\right )-2 (3 b+2 c) \text {arctanh}\left (\frac {x^2 (3 b+2 c)}{3 \sqrt {b^2 x^4-4 a c x y(x)}}\right )+(3 b+2 c) \log \left (9 a y(x)+3 b x^3+c x^3\right )}{6 (3 b+c)}+\frac {b \log (6 b y(x)+2 c y(x))}{2 (3 b+c)}&=c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**2 + b*x**2*Derivative(y(x), x) + c*x*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out