Internal problem ID [2822]
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 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=4 \,{\mathrm e}^{2 x} \ln \left (x \right )} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve([diff(y(x),x$2)-4*diff(y(x),x)+4*y(x)=4*exp(2*x)*ln(x),exp(2*x)],y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{2 x} x c_{1} +{\mathrm e}^{2 x} x^{2} \left (2 \ln \left (x \right )-3\right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 30
DSolve[y''[x]-4*y'[x]+4*y[x]==4*Exp[2*x]*Log[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 x} \left (-3 x^2+2 x^2 \log (x)+c_2 x+c_1\right ) \]