76.12.26 problem 38

Internal problem ID [17582]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 38
Date solved : Tuesday, January 28, 2025 at 10:44:29 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-\frac {a \,x^{2}}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.053 (sec). Leaf size: 57

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

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