Internal problem ID [1129]
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: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+y a^{2}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= x^{a} \end {align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 17
dsolve([x^2*diff(y(x),x$2)-(2*a-1)*x*diff(y(x),x)+a^2*y(x)=0,x^a],y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{a}+c_{2} x^{a} \ln \relax (x ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 18
DSolve[x^2*y''[x]-(2*a-1)*x*y'[x]+a^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^a (a c_2 \log (x)+c_1) \\ \end{align*}