65.10.5 problem 17.5

Internal problem ID [13788]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 17, Reduction of order. Exercises page 162
Problem number : 17.5
Date solved : Tuesday, January 28, 2025 at 06:03:28 AM
CAS classification : [_Hermite]

\begin{align*} y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 36

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

\[ y = c_{2} {\mathrm e}^{\frac {x^{2}}{2}}+\frac {x \left (i c_{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+2 c_{1} \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 61

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

\[ y(x)\to -\sqrt {\frac {\pi }{2}} c_2 \sqrt {x^2} \text {erfi}\left (\frac {\sqrt {x^2}}{\sqrt {2}}\right )+c_2 e^{\frac {x^2}{2}}+\sqrt {2} c_1 x \]