Internal problem ID [1132]
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: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \sin \relax (x ) y^{\prime \prime }-4 x \left (x \cos \relax (x )+\sin \relax (x )\right ) y^{\prime }+\left (2 x \cos \relax (x )+3 \sin \relax (x )\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 17
dsolve([4*x^2*sin(x)*diff(y(x),x$2)-4*x*(x*cos(x)+sin(x))*diff(y(x),x)+(2*x*cos(x)+3*sin(x))*y(x)=0,sqrt(x)],y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}+c_{2} \sqrt {x}\, \cos \relax (x ) \]
✓ Solution by Mathematica
Time used: 0.132 (sec). Leaf size: 21
DSolve[4*x^2*Sin[x]*y''[x]-4*x*(x*Cos[x]+Sin[x])*y'[x]+(2*x*Cos[x]+3*Sin[x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {\text {ArcCos}(\cos (x))} (c_2 \cos (x)+c_1) \\ \end{align*}