12.20.8 problem section 9.4, problem 23

Internal problem ID [2229]
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 23
Date solved : Tuesday, March 04, 2025 at 01:52:06 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-5 x^{2} y^{\prime \prime }+14 x y^{\prime }-18 y&=x^{3} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1\\ y^{\prime \prime }\left (1\right )&=7 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 25

ode:=x^3*diff(diff(diff(y(x),x),x),x)-5*x^2*diff(diff(y(x),x),x)+14*x*diff(y(x),x)-18*y(x) = x^3; 
ic:=y(1) = 0, D(y)(1) = 1, (D@@2)(y)(1) = 7; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 28

ode=x^3*D[y[x],{x,3}]-5*x^2*D[y[x],{x,2}]+14*x*D[y[x],x]-18*y[x]==x^3; 
ic={y[1]==0,Derivative[1][y][1]==1,Derivative[2][y][1]==7}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} x^2 \left (-2 x+x \log ^2(x)+4 x \log (x)+2\right ) \]

✓ Sympy. Time used: 0.347 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) - x**3 - 5*x**2*Derivative(y(x), (x, 2)) + 14*x*Derivative(y(x), x) - 18*y(x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 1, Subs(Derivative(y(x), (x, 2)), x, 1): 7} 
dsolve(ode,func=y(x),ics=ics)
 

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