60.3.374 problem 1391

Internal problem ID [11370]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1391
Date solved : Sunday, March 30, 2025 at 08:18:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x) = 1/x*(7*a*x^2+5)/(a*x^2+1)*diff(y(x),x)-(15*a*x^2+5)/x^2/(a*x^2+1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{5}+2 c_2 a \,x^{3}+c_2 x \]

✓ Mathematica. Time used: 0.348 (sec). Leaf size: 122

ode=D[y[x],{x,2}] == -(((5 + 15*a*x^2)*y[x])/(x^2*(1 + a*x^2))) + ((5 + 7*a*x^2)*D[y[x],x])/(x*(1 + a*x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3 a K[1]^2+5}{2 a K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^x-\frac {7 a K[2]^2+5}{a K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 a K[1]^2+5}{2 a K[1]^3+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (7*a*x**2 + 5)*Derivative(y(x), x)/(x*(a*x**2 + 1)) + (15*a*x**2 + 5)*y(x)/(x**2*(a*x**2 + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None