9.16 problem 16

Internal problem ID [1122]

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: 16.
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 }-4 x \left (x +1\right ) y^{\prime }+\left (2 x +3\right ) y-4 x^{\frac {5}{2}} {\mathrm e}^{2 x}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}

Solution by Maple

Time used: 0.017 (sec). Leaf size: 26

dsolve([4*x^2*diff(y(x),x$2)-4*x*(x+1)*diff(y(x),x)+(2*x+3)*y(x)=4*x^(5/2)*exp(2*x),x^(1/2)],y(x), singsol=all)
 

\[ y \relax (x ) = \sqrt {x}\, c_{2}+\sqrt {x}\, {\mathrm e}^{x} c_{1}+\frac {\sqrt {x}\, {\mathrm e}^{2 x}}{2} \]

Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 31

DSolve[4*x^2*y''[x]-4*x*(x+1)*y'[x]+(2*x+3)*y[x]==4*x^(5/2)*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \sqrt {x} \left (e^{2 x}+2 c_2 e^x+2 c_1\right ) \\ \end{align*}