54.3.315 problem 1332

Internal problem ID [12610]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1332
Date solved : Wednesday, October 01, 2025 at 02:15:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {y^{\prime }}{x +1}-\frac {\left (3 x +1\right ) y}{4 x^{2} \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

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

\[ y = \left (c_1 +c_2 \left (x +\ln \left (x \right )\right )\right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to \sqrt {x+1} \exp \left (\int _1^x\frac {1}{2 K[1]^2+2 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {1}{2 K[1]^2+2 K[1]}dK[1]\right )dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.340 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} + C_{2} \left (\log {\left (\frac {x}{x + 1} \right )} - \log {\left (\frac {x}{x + 1} - 1 \right )} - \frac {1}{\frac {x}{x + 1} - 1}\right )\right ) \left (x + 1\right ) {{}_{1}F_{0}\left (\begin {matrix} -1 \\ \end {matrix}\middle | {\frac {x}{x + 1}} \right )} \]