60.1.298 problem 304

Internal problem ID [10312]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 304
Date solved : Wednesday, March 05, 2025 at 10:10:54 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.553 (sec). Leaf size: 55

ode:=(10*x^3*y(x)^2+x^2*y(x)+2*x)*diff(y(x),x)+5*x^2*y(x)^3+x*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\tan \left (\operatorname {RootOf}\left (2 \sqrt {10}\, \ln \left (2\right )-\sqrt {10}\, \ln \left (5\right )+\sqrt {10}\, \ln \left (\frac {\tan \left (\textit {\_Z} \right )^{2} \sec \left (\textit {\_Z} \right )^{2}}{x^{2}}\right )+2 \sqrt {10}\, c_{1} +2 \textit {\_Z} \right )\right ) \sqrt {10}}{5 x} \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 402

ode=(10*x^3*y[x]^2+x^2*y[x]+2*x)*D[y[x],x]+5*x^2*y[x]^3+x*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (10 \exp \left (\int _1^{x K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) x^2 K[3]^2+\exp \left (\int _1^{x K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) x K[3]+2 \exp \left (\int _1^{x K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right )-\int _1^x\left (\frac {5 \exp \left (\int _1^{K[2] K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) K[2]^2 (1-5 K[2] K[3]) K[3]^3}{5 K[2]^2 K[3]^2+2}+15 \exp \left (\int _1^{K[2] K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) K[2] K[3]^2+\frac {\exp \left (\int _1^{K[2] K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) K[2] (1-5 K[2] K[3]) K[3]^2}{5 K[2]^2 K[3]^2+2}+2 \exp \left (\int _1^{K[2] K[3]}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) K[3]\right )dK[2]\right )dK[3]+\int _1^x\left (5 \exp \left (\int _1^{K[2] y(x)}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) K[2] y(x)^3+\exp \left (\int _1^{K[2] y(x)}\frac {1-5 K[1]}{5 K[1]^2+2}dK[1]\right ) y(x)^2\right )dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.181 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*x**2*y(x)**3 + x*y(x)**2 + (10*x**3*y(x)**2 + x**2*y(x) + 2*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \log {\left (x \right )} + \log {\left (x y{\left (x \right )} \right )} + \frac {\log {\left (x^{2} y^{2}{\left (x \right )} + \frac {2}{5} \right )}}{2} + \frac {\sqrt {10} \operatorname {atan}{\left (\frac {\sqrt {10} x y{\left (x \right )}}{2} \right )}}{10} = C_{1} \]