54.2.356 problem 935

Internal problem ID [12230]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 935
Date solved : Wednesday, October 01, 2025 at 01:15:19 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

\begin{align*} y^{\prime }&=-\frac {x}{2}+1+y^{2}+\frac {7 x^{2} y}{2}-2 x y+\frac {13 x^{4}}{16}-\frac {3 x^{3}}{2}+x^{2}+y^{3}+\frac {3 x^{2} y^{2}}{4}-3 x y^{2}+\frac {3 y x^{4}}{16}-\frac {3 x^{3} y}{2}+\frac {x^{6}}{64}-\frac {3 x^{5}}{16} \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 55

ode:=diff(y(x),x) = -1/2*x+1+y(x)^2+7/2*x^2*y(x)-2*x*y(x)+13/16*x^4-3/2*x^3+x^2+y(x)^3+3/4*x^2*y(x)^2-3*x*y(x)^2+3/16*y(x)*x^4-3/2*x^3*y(x)+1/64*x^6-3/16*x^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-4\right ) {\mathrm e}^{\textit {\_Z}}+c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}-4 \ln \left ({\mathrm e}^{\textit {\_Z}}-4\right )-4 c_1 +4 \textit {\_Z} -4 x +4\right )}}{4}-1-\frac {x^{2}}{4}+x \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 65

ode=D[y[x],x] == 1 - x/2 + x^2 - (3*x^3)/2 + (13*x^4)/16 - (3*x^5)/16 + x^6/64 - 2*x*y[x] + (7*x^2*y[x])/2 - (3*x^3*y[x])/2 + (3*x^4*y[x])/16 + y[x]^2 - 3*x*y[x]^2 + (3*x^2*y[x]^2)/4 + y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {1}{4} \left (3 x^2-12 x+4\right )+3 y(x)}{\sqrt [3]{2}}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]=\frac {1}{9} 2^{2/3} x+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.317 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**6/64 + 3*x**5/16 - 3*x**4*y(x)/16 - 13*x**4/16 + 3*x**3*y(x)/2 + 3*x**3/2 - 3*x**2*y(x)**2/4 - 7*x**2*y(x)/2 - x**2 + 3*x*y(x)**2 + 2*x*y(x) + x/2 - y(x)**3 - y(x)**2 + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x + \log {\left (\frac {x^{2}}{4} - x + y{\left (x \right )} \right )} - \log {\left (\frac {x^{2}}{4} - x + y{\left (x \right )} + 1 \right )} + \frac {1}{\frac {x^{2}}{4} - x + y{\left (x \right )}} = 0 \]