6.20.7 problem section 9.4, problem 22

Internal problem ID [2228]
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 22
Date solved : Tuesday, September 30, 2025 at 05:25:14 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.207 (sec). Leaf size: 19

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

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