6.20.13 problem section 9.4, problem 33

Internal problem ID [2234]
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 33
Date solved : Tuesday, September 30, 2025 at 05:25:17 AM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+5 x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }-6 x y^{\prime }+6 y&=40 x^{3} \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=-1 \\ y^{\prime }\left (-1\right )&=-7 \\ y^{\prime \prime }\left (-1\right )&=-1 \\ y^{\prime \prime \prime }\left (-1\right )&=-31 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 29

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+5*x^3*diff(diff(diff(y(x),x),x),x)-3*x^2*diff(diff(y(x),x),x)-6*x*diff(y(x),x)+6*y(x) = 40*x^3; 
ic:=[y(-1) = -1, D(y)(-1) = -7, (D@@2)(y)(-1) = -1, (D@@3)(y)(-1) = -31]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\ln \left (x \right ) x^{5}-1+\left (-i \pi -2\right ) x^{5}+x^{3}+x}{x^{2}} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 32

ode=x^4*D[y[x],{x,4}]+5*x^3*D[y[x],{x,3}]-3*x^2*D[y[x],{x,2}]-6*x*D[y[x],x]+6*y[x]==40*x^3; 
ic={y[-1]==-1,Derivative[1][y][-1]==-7,Derivative[2][y][-1]==-1,Derivative[3][y][-1]==-31}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(-2-i \pi ) x^5+x^5 \log (x)+x^3+x-1}{x^2} \end{align*}

✓ Sympy. Time used: 0.282 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 5*x**3*Derivative(y(x), (x, 3)) - 40*x**3 - 3*x**2*Derivative(y(x), (x, 2)) - 6*x*Derivative(y(x), x) + 6*y(x),0) 
ics = {y(-1): -1, Subs(Derivative(y(x), x), x, -1): -7, Subs(Derivative(y(x), (x, 2)), x, -1): -1, Subs(Derivative(y(x), (x, 3)), x, -1): -31} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{3} \log {\left (x \right )} + x^{3} \left (-2 - i \pi \right ) + x + \frac {1}{x} - \frac {1}{x^{2}} \]