Internal problem ID [13228]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.2 (m).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {x}{2}-\frac {1}{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 15
dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,sinh(ln(x))],y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sin \left (\ln \left (x \right )\right )+c_{2} \cos \left (\ln \left (x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 18
DSolve[x^2*y''[x]+x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cos (\log (x))+c_2 \sin (\log (x)) \]