49.13.6 problem 1(f)

Internal problem ID [7682]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 121
Problem number : 1(f)
Date solved : Monday, January 27, 2025 at 03:09:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+2 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: 0.005 (sec). Leaf size: 23

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

\[ y = {\mathrm e}^{x^{2}} c_{2} +x \left (-\operatorname {erfi}\left (x \right ) c_{2} \sqrt {\pi }+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 43

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

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