54.2.71 problem 647
Internal
problem
ID
[11945]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
647
Date
solved
:
Sunday, October 12, 2025 at 02:05:22 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=\frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y} \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 318
ode:=diff(y(x),x) = (a*y(x)^2+b*x^2)^2*x/a^(5/2)/y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\mathrm e}^{-\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}} \sqrt {-\left (c_1 \left (\sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right ) {\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+b \,x^{2}-\sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}\right ) \left (c_1 \,{\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+1\right ) {\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right )}{a^{{3}/{2}}}} a}}{a \left (c_1 \,{\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+1\right )} \\
y &= -\frac {{\mathrm e}^{-\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}} \sqrt {-\left (c_1 \left (\sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right ) {\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+b \,x^{2}-\sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}\right ) \left (c_1 \,{\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+1\right ) {\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{{3}/{2}}}}\, a^{{3}/{2}}+b \,x^{2}\right )}{a^{{3}/{2}}}} a}}{a \left (c_1 \,{\mathrm e}^{2 x^{2} \sqrt {-\frac {b}{a^{{3}/{2}}}}}+1\right )} \\
\end{align*}
✓ Mathematica. Time used: 0.22 (sec). Leaf size: 201
ode=D[y[x],x] == (x*(b*x^2 + a*y[x]^2)^2)/(a^(5/2)*y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\left (\frac {a^3 b^2 K[1]}{b^2 K[1]^4+2 a b y(x)^2 K[1]^2+a^2 y(x)^4+a^{3/2} b}-a^{3/2} b K[1]\right )dK[1]+\int _1^{y(x)}\left (\frac {a^4 b K[2]}{b^2 x^4+2 a b K[2]^2 x^2+a^2 K[2]^4+a^{3/2} b}-\int _1^x-\frac {a^3 b^2 K[1] \left (4 a^2 K[2]^3+4 a b K[1]^2 K[2]\right )}{\left (b^2 K[1]^4+2 a b K[2]^2 K[1]^2+a^2 K[2]^4+a^{3/2} b\right )^2}dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(Derivative(y(x), x) - x*(a*y(x)**2 + b*x**2)**2/(a**(5/2)*y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - x*(a**2*y(x)**4 + 2*a*b*x**2*y(x)**2 + b**2*x**4)/(a**(5/2)*y(x)) cannot be solved by the factorable group method