82.54.8 problem Ex. 8

Internal problem ID [19024]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 8
Date solved : Tuesday, January 28, 2025 at 12:46:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 x -1\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: 0.006 (sec). Leaf size: 13

dsolve([x*diff(y(x),x$2)-(2*x-1)*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: 0.021 (sec). Leaf size: 17

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

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