12.20.17 problem section 9.4, problem 41

Internal problem ID [2238]
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 41
Date solved : Tuesday, March 04, 2025 at 01:52:14 PM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y&=F \left (x \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 36

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+2*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+4*y(x) = F(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_4 +\int \left (2 c_2 x +c_3 +\int \left (\int \frac {c_1 +\int F \left (x \right )d x}{x^{4}}d x \right )d x \right ) x^{2}d x}{x^{2}} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 104

ode=x^4*D[y[x],{x,4}]+6*x^3*D[y[x],{x,3}]+2*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+4*y[x]==f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^4 \int _1^x\frac {f(K[4])}{12 K[4]^3}dK[4]+x^3 \int _1^x-\frac {f(K[3])}{6 K[3]^2}dK[3]+x \int _1^x\frac {1}{6} f(K[2])dK[2]+\int _1^x-\frac {1}{12} f(K[1]) K[1]dK[1]+c_4 x^4+c_3 x^3+c_2 x+c_1}{x^2} \]

✓ Sympy. Time used: 1.302 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + 2*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) - F(x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2}} + \frac {C_{2}}{x} + C_{3} x + C_{4} x^{2} + \frac {x^{2} \int \frac {F{\left (x \right )}}{x^{3}}\, dx}{12} - \frac {x \int \frac {F{\left (x \right )}}{x^{2}}\, dx}{6} + \frac {\int F{\left (x \right )}\, dx}{6 x} - \frac {\int x F{\left (x \right )}\, dx}{12 x^{2}} \]