6.20.9 problem section 9.4, problem 25

Internal problem ID [2230]
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 25
Date solved : Tuesday, September 30, 2025 at 05:25:15 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+16 x y^{\prime }-16 y&=9 x^{4} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=1 \\ y^{\prime \prime }\left (1\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 29

ode:=x^3*diff(diff(diff(y(x),x),x),x)-6*x^2*diff(diff(y(x),x),x)+16*x*diff(y(x),x)-16*y(x) = 9*x^4; 
ic:=[y(1) = 2, D(y)(1) = 1, (D@@2)(y)(1) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -x^{4}+\frac {3 \ln \left (x \right )^{2} x^{4}}{2}+2 x^{4} \ln \left (x \right )+3 x \]

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

ode=x^3*D[y[x],{x,3}]-6*x^2*D[y[x],{x,2}]+16*x*D[y[x],x]-16*y[x]==9*x^4; 
ic={y[1]==2,Derivative[1][y][1]==1,Derivative[2][y][1]==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x^4+\frac {3}{2} x^4 \log ^2(x)+2 x^4 \log (x)+3 x \end{align*}

✓ Sympy. Time used: 0.241 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*x**4 + x**3*Derivative(y(x), (x, 3)) - 6*x**2*Derivative(y(x), (x, 2)) + 16*x*Derivative(y(x), x) - 16*y(x),0) 
ics = {y(1): 2, Subs(Derivative(y(x), x), x, 1): 1, Subs(Derivative(y(x), (x, 2)), x, 1): 5} 
dsolve(ode,func=y(x),ics=ics)
 

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