83.47.5 problem Ex 5 page 82

Internal problem ID [19503]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VI. Homogeneous linear equations with variable coefficients
Problem number : Ex 5 page 82
Date solved : Thursday, March 13, 2025 at 02:42:58 PM
CAS classification : [[_3rd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 36

ode:=x^3*diff(diff(diff(y(x),x),x),x)+3*x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {c_{2} x^{{3}/{2}} \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+c_3 \,x^{{3}/{2}} \cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+c_{1}}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode=x^3*D[y[x],{x,3}]+3*x^2*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_3 x^{3/2} \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+c_2 x^{3/2} \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )+c_1}{x} \]

✓ Sympy. Time used: 0.233 (sec). Leaf size: 37

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

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