54.2.3 problem 579
Internal
problem
ID
[11877]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
579
Date
solved
:
Tuesday, September 30, 2025 at 11:38:25 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{\prime }&=-\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right ) \end{align*}
✓ Maple. Time used: 0.015 (sec). Leaf size: 35
ode:=diff(y(x),x) = -1/2*a*x+F(y(x)+1/4*x^2*a+1/2*b*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {a \,x^{2}}{4}-\frac {b x}{2}+\operatorname {RootOf}\left (-x +2 \int _{}^{\textit {\_Z}}\frac {1}{2 F \left (\textit {\_a} \right )+b}d \textit {\_a} +c_1 \right )
\]
✓ Mathematica. Time used: 0.12 (sec). Leaf size: 514
ode=D[y[x],x] == -1/2*(a*x) + F[(b*x)/2 + (a*x^2)/4 + y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}-\frac {b \int _1^x\left (\frac {2 a K[1] F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right ) \int _1^x\left (\frac {2 a K[1] F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2}{b+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right )}dK[2]+\int _1^x\left (\frac {2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}-\frac {a K[1]}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}\right )dK[1]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
F = Function("F")
ode = Eq(a*x/2 - F(a*x**2/4 + b*x/2 + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a*x/2 - F(a*x**2/4 + b*x/2 + y(x)) + Derivative(y(x), x) cannot be solved by the lie group method