67.8.40 problem 13.6 (f)

Internal problem ID [16535]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.6 (f)
Date solved : Thursday, October 02, 2025 at 01:36:09 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} x y^{\prime \prime \prime }+2 y^{\prime \prime }&=6 x \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 18

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

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

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

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

\begin{align*} y(x)&\to \frac {1}{3} \left (x^3+6 x-6 \log (x)-1\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 19

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

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