9.35 problem 35

Internal problem ID [1141]

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: 35.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _exact, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+4 y^{\prime } x +2 y-2-x=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{-2+x} \end {align*}

With initial conditions \begin {align*} \left [y \relax (0) = -{\frac {1}{3}}, y^{\prime }\relax (0) = -1\right ] \end {align*}

Solution by Maple

Time used: 0.013 (sec). Leaf size: 27

dsolve([(x^2-4)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+2*y(x) = 2+x, 1/(-2+x), y(0) = -1/3, D(y)(0) = -1],y(x), singsol=all)
 

\[ y \relax (x ) = \frac {x^{3}+6 x^{2}+24 x +8}{6 x^{2}-24} \]

Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 27

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

\begin{align*} y(x)\to \frac {2 (x-3) x (x+9)-5}{12 \left (x^2-4\right )} \\ \end{align*}