9.1.7 problem problem 44

Internal problem ID [934]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.2, Higher-Order Linear Differential Equations. General solutions of Linear Equations. Page 288
Problem number : problem 44
Date solved : Monday, January 27, 2025 at 03:22:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\cos \left (x \right )}{\sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 17

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

\[ y = \frac {c_1 \sin \left (x \right )+c_2 \cos \left (x \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 33

DSolve[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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