12.20.11 problem section 9.4, problem 30

Internal problem ID [2232]
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 30
Date solved : Tuesday, March 04, 2025 at 01:52:08 PM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+3 x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y&=9 x^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-7\\ y^{\prime }\left (1\right )&=-11\\ y^{\prime \prime }\left (1\right )&=-5\\ y^{\prime \prime \prime }\left (1\right )&=6 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 14

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+3*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) = 9*x^2; 
ic:=y(1) = -7, D(y)(1) = -11, (D@@2)(y)(1) = -5, (D@@3)(y)(1) = 6; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = x^{2} \left (-7+3 \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 15

ode=x^4*D[y[x],{x,4}]+3*x^3*D[y[x],{x,3}]-x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-2*y[x]==9*x^2; 
ic={y[1]==-7,Derivative[1][y][1]==-11,Derivative[2][y][1]==-5,Derivative[3][y][1]==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^2 (3 \log (x)-7) \]

✓ Sympy. Time used: 0.454 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 3*x**3*Derivative(y(x), (x, 3)) - x**2*Derivative(y(x), (x, 2)) - 9*x**2 + 2*x*Derivative(y(x), x) - 2*y(x),0) 
ics = {y(1): -7, Subs(Derivative(y(x), x), x, 1): -11, Subs(Derivative(y(x), (x, 2)), x, 1): -5, Subs(Derivative(y(x), (x, 3)), x, 1): 6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (3 x \log {\left (x \right )} - 7 x\right ) \]