Internal problem ID [2819]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page
572
Problem number: Problem 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+2 y=8 x^{2} {\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve([x*diff(y(x),x$2)-(2*x+1)*diff(y(x),x)+2*y(x)=8*x^2*exp(2*x),exp(2*x)],y(x), singsol=all)
\[ y \left (x \right ) = \left (1+2 x \right ) c_{2} +c_{1} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} x^{2} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 32
DSolve[x*y''[x]-(2*x+1)*y'[x]+2*y[x]==8*x^2*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 x} \left (2 x^2-1+c_1\right )-\frac {1}{4} c_2 (2 x+1) \]