Internal problem ID [5258]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x^{2}} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 18
dsolve([diff(y(x),x$2)-4*x*diff(y(x),x)+(4*x^2-2)*y(x)=0,exp(x^2)],y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{x^{2}}+c_{2} {\mathrm e}^{x^{2}} x \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 18
DSolve[y''[x]-4*x*y'[x]+(4*x^2-2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{x^2} (c_2 x+c_1) \\ \end{align*}