12.9.9 problem 9

Internal problem ID [1765]
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 : 9
Date solved : Monday, January 27, 2025 at 05:34:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-4 y&=-6 x -4 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 19

dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)-4*y(x)=-6*x-4,x^2],singsol=all)
 

\[ y = \frac {c_2}{x^{2}}+c_1 \,x^{2}+2 x +1 \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 22

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]-4*y[x]==-6*x-4,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 x^2+\frac {c_1}{x^2}+2 x+1 \]