Internal problem ID [5587]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN
SOLUTION TO FIND ANOTHER. Page 74
Problem number: 3.
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 {x^{2} y^{\prime \prime }+x y^{\prime }-4 y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 15
dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)-4*y(x)=0,x^2],y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{2}+\frac {c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 18
DSolve[x^2*y''[x]+x*y'[x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x^4+c_1}{x^2} \\ \end{align*}