Internal problem ID [1112]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page
253
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y-4 \sqrt {x}\, {\mathrm e}^{x} \left (4 x +1\right )=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x}\, {\mathrm e}^{x} \end {align*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 32
dsolve([4*x^2*diff(y(x),x$2)+(4*x-8*x^2)*diff(y(x),x)+(4*x^2-4*x-1)*y(x)=4*x^(1/2)*exp(x)*(1+4*x),x^(1/2)*exp(x)],y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{x} c_{2}}{\sqrt {x}}+\sqrt {x}\, {\mathrm e}^{x} c_{1}+{\mathrm e}^{x} \sqrt {x}\, \left (\ln \relax (x )+2 x -1\right ) \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 30
DSolve[4*x^2*y''[x]+(4*x-8*x^2)*y'[x]+(4*x^2-4*x-1)*y[x]==4*x^(1/2)*Exp[x]*(1+4*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^x (x \log (x)+x (2 x-1+c_2)+c_1)}{\sqrt {x}} \\ \end{align*}