Internal problem ID [10704]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Hermite]
\[ \boxed {y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve([diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,x],y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x +c_{2} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 47
DSolve[y''[x]-x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (2 c_1-\sqrt {\pi } c_2 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )\right )}{\sqrt {2}}+c_2 e^{\frac {x^2}{2}} \\ \end{align*}