6.20.12 problem section 9.4, problem 32

Internal problem ID [2233]
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 32
Date solved : Tuesday, September 30, 2025 at 05:25:17 AM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{4} y^{\prime \prime \prime \prime }+24 x^{3} y^{\prime \prime \prime }+23 x^{2} y^{\prime \prime }-x y^{\prime }+y&=6 x \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=0 \\ y^{\prime \prime }\left (1\right )&=4 \\ y^{\prime \prime \prime }\left (1\right )&=-{\frac {37}{4}} \\ \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 27

ode:=4*x^4*diff(diff(diff(diff(y(x),x),x),x),x)+24*x^3*diff(diff(diff(y(x),x),x),x)+23*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 6*x; 
ic:=[y(1) = 2, D(y)(1) = 0, (D@@2)(y)(1) = 4, (D@@3)(y)(1) = -37/4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 26

ode=4*x^4*D[y[x],{x,4}]+24*x^3*D[y[x],{x,3}]+23*x^2*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==6*x; 
ic={y[1]==2,Derivative[1][y][1]==0,Derivative[2][y][1]==4,Derivative[3][y][1]==-37/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x-\sqrt {x}+\frac {1}{\sqrt {x}}+\frac {1}{x}+x \log (x) \end{align*}

✓ Sympy. Time used: 0.255 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**4*Derivative(y(x), (x, 4)) + 24*x**3*Derivative(y(x), (x, 3)) + 23*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) - 6*x + y(x),0) 
ics = {y(1): 2, Subs(Derivative(y(x), x), x, 1): 0, Subs(Derivative(y(x), (x, 2)), x, 1): 4, Subs(Derivative(y(x), (x, 3)), x, 1): -37/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \sqrt {x} + x \log {\left (x \right )} + x + \frac {1}{x} + \frac {1}{\sqrt {x}} \]