Internal problem ID [284]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \relax (x )}{\sqrt {x}} \end {align*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 19
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)],y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sin \relax (x )}{\sqrt {x}}+\frac {c_{2} \cos \relax (x )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 19
DSolve[(1-x^2)*y''[x]-2*x*y'[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+c_2 \left (x \tanh ^{-1}(x)-1\right ) \\ \end{align*}