54.2.263 problem 841

Internal problem ID [12137]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 841
Date solved : Sunday, October 12, 2025 at 02:22:18 AM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 97

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

\begin{align*} y &= -\frac {2 \sqrt {a^{{3}/{2}} \left (c_1 x +1\right ) \left (\left (c_1 x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right )}}{a^{{3}/{2}} \left (2 c_1 x +2\right )} \\ y &= \frac {\sqrt {a^{{3}/{2}} \left (c_1 x +1\right ) \left (\left (c_1 x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right )}}{a^{{3}/{2}} \left (c_1 x +1\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.568 (sec). Leaf size: 431

ode=D[y[x],x] == (Sqrt[a]*c^2 - 2*Sqrt[a]*b*c*x^2 + b*x^3 + Sqrt[a]*b^2*x^4 + 2*a^(3/2)*c*y[x]^2 - 2*a^(3/2)*b*x^2*y[x]^2 + a^(5/2)*y[x]^4)/(a*x^2*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {4 a^4 x^2 K[2]}{\left (-2 \sqrt {a} b x^2-x+2 a^{3/2} K[2]^2+2 \sqrt {a} c\right )^2}-\int _1^x\left (\frac {2 K[2] a^{7/2}}{b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^2}+\frac {4 K[2] \left (-8 b K[1] K[2]^2 a^4+2 K[2]^2 a^{7/2}-8 b c K[1] a^3+2 c a^{5/2}-K[1] a^2\right ) a^{3/2}}{b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^3}+\frac {4 a^{7/2} K[2]-16 a^4 b K[1] K[2]}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (-a^{5/2}+\frac {a^2}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} y(x)^2-2 \sqrt {a} c\right )}+\frac {-8 b K[1] y(x)^2 a^4+2 y(x)^2 a^{7/2}-8 b c K[1] a^3+2 c a^{5/2}-K[1] a^2}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} y(x)^2-2 \sqrt {a} c\right )^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (a**(5/2)*y(x)**4 - 2*a**(3/2)*b*x**2*y(x)**2 + 2*a**(3/2)*c*y(x)**2 + sqrt(a)*b**2*x**4 - 2*sqrt(a)*b*c*x**2 + sqrt(a)*c**2 + b*x**3)/(a*x**2*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out