59.1.451 problem 465

Internal problem ID [9623]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 465
Date solved : Sunday, March 30, 2025 at 02:38:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 17

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

\[ y = \left (x +1\right ) \left (c_1 \sinh \left (x \right )+c_2 \cosh \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.246 (sec). Leaf size: 146

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (1-\sqrt {2}+i \sqrt {7}\right ),1+i \sqrt {7},2 \sqrt {2} (x+1)\right )+c_2 L_{\frac {1}{2} \left (-1+\sqrt {2}-i \sqrt {7}\right )}^{i \sqrt {7}}\left (2 \sqrt {2} (x+1)\right )\right ) \exp \left (\int _1^x\frac {-2 \sqrt {2} K[1]+2 K[1]+i \sqrt {7}-2 \sqrt {2}+1}{2 K[1]+2}dK[1]\right ) \]

✓ Sympy. Time used: 0.306 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = \left (x + 1\right )^{\frac {3}{2}} \left (C_{1} J_{\frac {1}{2}}\left (i \left (x + 1\right )\right ) + C_{2} Y_{\frac {1}{2}}\left (i \left (x + 1\right )\right )\right ) \]