87.14.28 problem 28

Internal problem ID [23547]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 28
Date solved : Thursday, October 02, 2025 at 09:42:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \,{\mathrm e} \\ y^{\prime }\left (1\right )&=-3 \,{\mathrm e} \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 13

ode:=x*diff(diff(y(x),x),x)+(1-2*x)*diff(y(x),x)+(x-1)*y(x) = 0; 
ic:=[y(1) = 2*exp(1), D(y)(1) = -3*exp(1)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (2-5 \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 15

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

\begin{align*} y(x)&\to e^x (2-5 \log (x)) \end{align*}

✗ Sympy

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

ValueError : Couldnt solve for initial conditions