12.20.15 problem section 9.4, problem 36

Internal problem ID [2236]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.4. Variation of Parameters for Higher Order Equations. Page 503
Problem number : section 9.4, problem 36
Date solved : Saturday, March 29, 2025 at 11:51:32 PM
CAS classification : [[_3rd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=F \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

\[ y = \left (c_3 +\int \frac {c_2 +\int \frac {c_1 +\int F \left (x \right )d x}{x^{3}}d x}{x^{2}}d x \right ) x^{2} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 82

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

\[ y(x)\to \frac {x^3 \int _1^x\frac {f(K[3])}{3 K[3]^3}dK[3]+x^2 \int _1^x-\frac {f(K[2])}{2 K[2]^2}dK[2]+\int _1^x\frac {1}{6} f(K[1])dK[1]+c_3 x^3+c_2 x^2+c_1}{x} \]

✓ Sympy. Time used: 1.029 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} + C_{2} x + C_{3} x^{2} + \frac {x^{2} \int \frac {F{\left (x \right )}}{x^{3}}\, dx}{3} - \frac {x \int \frac {F{\left (x \right )}}{x^{2}}\, dx}{2} + \frac {\int F{\left (x \right )}\, dx}{6 x} \]