Internal problem ID [13221]
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 (f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve([diff(y(x),x$2)-(4+2/x)*diff(y(x),x)+(4+4/x)*y(x)=0,exp(2*x)],y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{2 x} x^{3} \]
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 25
DSolve[y''[x]-(4+2/x)*y'[x]+(4+4/x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{3} e^{2 x} \left (c_2 x^3+3 c_1\right ) \]