81.1.7 problem 6

Internal problem ID [18531]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter I. Introduction. Exercises at page 13
Problem number : 6
Date solved : Monday, March 31, 2025 at 05:41:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 17

ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x) = x*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \sinh \left (x \right )+c_2 \cosh \left (x \right )}{x} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 28

ode=x*D[y[x],{x,2}]+2*D[y[x],x]==x*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.215 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (i x\right ) + C_{2} Y_{\frac {1}{2}}\left (i x\right )}{\sqrt {x}} \]