20.11.2 problem Problem 2

Internal problem ID [3784]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 2
Date solved : Monday, January 27, 2025 at 08:02:16 AM
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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 13

dsolve([x*diff(y(x),x$2)+(1-2*x)*diff(y(x),x)+(x-1)*y(x)=0,exp(x)],singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} \ln \left (x \right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 17

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

\[ y(x)\to e^x (c_2 \log (x)+c_1) \]