62.28.2 problem Ex 2

Internal problem ID [12870]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 51. Cauchy linear equation. Page 114
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:49:00 PM
CAS classification : [[_3rd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+2 y&=10 x +\frac {10}{x} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 39

ode:=x^3*diff(diff(diff(y(x),x),x),x)+2*x^2*diff(diff(y(x),x),x)+2*y(x) = 10*x+10/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {5 \sin \left (\ln \left (x \right )\right ) x^{2} c_3 +5 \cos \left (\ln \left (x \right )\right ) x^{2} c_{2} +25 x^{2}+10 \ln \left (x \right )+c_{1} +8}{5 x} \]

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 106

ode=x^3*D[y[x],{x,3}]+2*x^2*D[y[x],{x,2}]+2*y[x]==10*(x+1/x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x \sin (\log (x)) \int _1^x\frac {2 \left (K[1]^2+1\right ) (2 \cos (\log (K[1]))-\sin (\log (K[1])))}{K[1]^3}dK[1]+x \cos (\log (x)) \int _1^x-\frac {2 \left (K[2]^2+1\right ) (\cos (\log (K[2]))+2 \sin (\log (K[2])))}{K[2]^3}dK[2]+x+\frac {2 \log (x)}{x}+\frac {c_3}{x}+c_2 x \cos (\log (x))+c_1 x \sin (\log (x)) \]

✓ Sympy. Time used: 0.448 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) + 2*x**2*Derivative(y(x), (x, 2)) - 10*x + 2*y(x) - 10/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + x^{2} \left (C_{2} \sin {\left (\log {\left (x \right )} \right )} + C_{3} \cos {\left (\log {\left (x \right )} \right )} + 5\right ) + 2 \log {\left (x \right )}}{x} \]