74.15.37 problem 37

Internal problem ID [16397]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 37
Date solved : Thursday, March 13, 2025 at 08:12:28 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+5 x y^{\prime }-5 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 24

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

\[ y(x)\to -\frac {1}{2} x (-x \sin (\log (x))+3 x \cos (\log (x))-3) \]

✓ Sympy. Time used: 0.271 (sec). Leaf size: 24

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)) + 5*x*Derivative(y(x), x) - 5*y(x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): -1, Subs(Derivative(y(x), (x, 2)), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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