83.17.14 problem 14

Internal problem ID [19139]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter III. Ordinary linear differential equations with constant coefficients. Misc. Examples on chapter III at page 50
Problem number : 14
Date solved : Monday, March 31, 2025 at 06:49:33 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y&=24 x \cos \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=12 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 18

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(y(x),x),x)+y(x) = 24*x*cos(x); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 12; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 19

ode=D[y[x],{x,4}]+2*D[y[x],{x,2}]+y[x]==24*x*Cos[x]; 
ic={y[0]==0,Derivative[1][y][0] == 0,Derivative[2][y][0] == 0,Derivative[3][y][0] == 12}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^2 (3 \sin (x)-x \cos (x)) \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-24*x*cos(x) + y(x) + 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0, Subs(Derivative(y(x), (x, 3)), x, 0): 12} 
dsolve(ode,func=y(x),ics=ics)
 

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