Internal problem ID [1127]
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: 21.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (\sqrt {x}\right ) \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve([4*x*diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0,sin(sqrt(x))],y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (\sqrt {x}\right )+c_{2} \cos \left (\sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 24
DSolve[4*x*y''[x]+2*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sqrt {x}\right )+c_2 \sin \left (\sqrt {x}\right ) \\ \end{align*}