78.11.4 problem 4

Internal problem ID [18275]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 16. The Use of a Known Solution to find Another. Problems at page 121
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 11:45:01 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 15

dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)-4*y(x)=0,x^2],singsol=all)
 

\[ y = \frac {c_{1} x^{4}+c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 18

DSolve[x^2*D[y[x],{x,2}] +x*D[y[x],x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_2 x^4+c_1}{x^2} \]