54.3.305 problem 1322

Internal problem ID [12600]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1322
Date solved : Wednesday, October 01, 2025 at 02:12:47 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }+2 x \left (3 x +2\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

ode:=x^2*(1+x)*diff(diff(y(x),x),x)+2*x*(2+3*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 +\left (-4 \ln \left (x \right )+4 \ln \left (x +1\right )-\frac {12 x^{3}+6 x^{2}-2 x +1}{3 x^{3} \left (x +1\right )}\right ) c_2 \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 45

ode=2*x*(2 + 3*x)*D[y[x],x] + x^2*(1 + x)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\exp \left (\int _1^{K[2]}-\frac {6 K[1]+4}{K[1]^2+K[1]}dK[1]\right ) c_1dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 0.245 (sec). Leaf size: 87

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + 2*x*(3*x + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + x^{\frac {- \left (\operatorname {re}{\left (x\right )} + 1\right ) \left (5 \operatorname {re}{\left (x\right )} + 3\right ) - 5 \left (\operatorname {im}{\left (x\right )}\right )^{2}}{\left (\operatorname {re}{\left (x\right )} + 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}}} \left (C_{2} \sin {\left (\frac {2 \log {\left (x \right )} \left |{\operatorname {im}{\left (x\right )}}\right |}{\left (\operatorname {re}{\left (x\right )} + 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}} \right )} + C_{3} \cos {\left (\frac {2 \log {\left (x \right )} \operatorname {im}{\left (x\right )}}{\left (\operatorname {re}{\left (x\right )} + 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}} \right )}\right ) \]