49.13.3 problem 1(c)

Internal problem ID [7679]
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(c)
Date solved : Monday, January 27, 2025 at 03:09:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 14

dsolve([diff(y(x),x$2)-4*x*diff(y(x),x)+(4*x^2-2)*y(x)=0,exp(x^2)],singsol=all)
 

\[ y = {\mathrm e}^{x^{2}} \left (c_{2} x +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 18

DSolve[D[y[x],{x,2}]-4*x*D[y[x],x]+(4*x^2-2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{x^2} (c_2 x+c_1) \]