60.2.372 problem 950

Internal problem ID [10946]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 950
Date solved : Sunday, March 30, 2025 at 07:28:37 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

\begin{align*} y^{\prime }&=-\frac {a x}{2}+1+y^{2}+\frac {a \,x^{2} y}{2}+b x y+\frac {a^{2} x^{4}}{16}+\frac {a \,x^{3} b}{4}+\frac {b^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} a y^{2}}{4}+\frac {3 y^{2} b x}{2}+\frac {3 y a^{2} x^{4}}{16}+\frac {3 y a \,x^{3} b}{4}+\frac {3 y b^{2} x^{2}}{4}+\frac {a^{3} x^{6}}{64}+\frac {3 a^{2} x^{5} b}{32}+\frac {3 a \,x^{4} b^{2}}{16}+\frac {b^{3} x^{3}}{8} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 42

ode:=diff(y(x),x) = -1/2*a*x+1+y(x)^2+1/2*a*x^2*y(x)+b*x*y(x)+1/16*a^2*x^4+1/4*a*x^3*b+1/4*b^2*x^2+y(x)^3+3/4*x^2*a*y(x)^2+3/2*y(x)^2*b*x+3/16*y(x)*a^2*x^4+3/4*y(x)*a*x^3*b+3/4*y(x)*b^2*x^2+1/64*a^3*x^6+3/32*a^2*x^5*b+3/16*a*x^4*b^2+1/8*b^3*x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {a \,x^{2}}{4}-\frac {b x}{2}+\operatorname {RootOf}\left (-x +2 \int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+b +2}d \textit {\_a} +c_1 \right ) \]

✓ Mathematica. Time used: 0.327 (sec). Leaf size: 94

ode=D[y[x],x] == 1 - (a*x)/2 + (b^2*x^2)/4 + (a*b*x^3)/4 + (b^3*x^3)/8 + (a^2*x^4)/16 + (3*a*b^2*x^4)/16 + (3*a^2*b*x^5)/32 + (a^3*x^6)/64 + b*x*y[x] + (a*x^2*y[x])/2 + (3*b^2*x^2*y[x])/4 + (3*a*b*x^3*y[x])/4 + (3*a^2*x^4*y[x])/16 + y[x]^2 + (3*b*x*y[x]^2)/2 + (3*a*x^2*y[x]^2)/4 + y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}}\frac {1}{K[1]^3-\frac {3\ 2^{2/3} K[1]}{(27 b+58)^{2/3}}+1}dK[1]=\frac {(27 b+58)^{2/3} x}{9\ 2^{2/3}}+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a**3*x**6/64 - 3*a**2*b*x**5/32 - 3*a**2*x**4*y(x)/16 - a**2*x**4/16 - 3*a*b**2*x**4/16 - 3*a*b*x**3*y(x)/4 - a*b*x**3/4 - 3*a*x**2*y(x)**2/4 - a*x**2*y(x)/2 + a*x/2 - b**3*x**3/8 - 3*b**2*x**2*y(x)/4 - b**2*x**2/4 - 3*b*x*y(x)**2/2 - b*x*y(x) - y(x)**3 - y(x)**2 + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a**3*x**6/64 - 3*a**2*b*x**5/32 - 3*a**2*x**4*y(x)/16 - a**2*x**4/16 - 3*a*b**2*x**4/16 - 3*a*b*x**3*y(x)/4 - a*b*x**3/4 - 3*a*x**2*y(x)**2/4 - a*x**2*y(x)/2 + a*x/2 - b**3*x**3/8 - 3*b**2*x**2*y(x)/4 - b**2*x**2/4 - 3*b*x*y(x)**2/2 - b*x*y(x) - y(x)**3 - y(x)**2 + Derivative(y(x), x) - 1 cannot be solved by the lie group method