60.2.373 problem 951

Internal problem ID [10947]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 951
Date solved : Wednesday, March 05, 2025 at 01:26:46 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.247 (sec). Leaf size: 93

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**3*x**3/8 - 3*a**2*x**4/16 - 3*a**2*x**2*y(x)/4 - a**2*x**2/4 - 3*a*x**5/32 - 3*a*x**3*y(x)/4 - a*x**3/4 - 3*a*x*y(x)**2/2 - a*x*y(x) - x**6/64 - 3*x**4*y(x)/16 - x**4/16 - 3*x**2*y(x)**2/4 - x**2*y(x)/2 + x/2 - 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**3/8 - 3*a**2*x**4/16 - 3*a**2*x**2*y(x)/4 - a**2*x**2/4 - 3*a*x**5/32 - 3*a*x**3*y(x)/4 - a*x**3/4 - 3*a*x*y(x)**2/2 - a*x*y(x) - x**6/64 - 3*x**4*y(x)/16 - x**4/16 - 3*x**2*y(x)**2/4 - x**2*y(x)/2 + x/2 - y(x)**3 - y(x)**2 + Derivative(y(x), x) - 1 cannot be solved by the lie group method