85.43.2 problem 2 (a)

Internal problem ID [22780]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. B Exercises at page 180
Problem number : 2 (a)
Date solved : Thursday, October 02, 2025 at 09:14:36 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} a^{2} b^{2} y+\left (a^{2}+b^{2}\right ) y^{\prime \prime }+y^{\prime \prime \prime \prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 34

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

\[ y = \frac {-b^{2} \cos \left (a x \right )+a^{2} \cos \left (b x \right )}{a^{2}-b^{2}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 35

ode=D[y[x],{x,4}]+(a^2+b^2)*D[y[x],{x,2}]+(a^2*b^2)*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {a^2 \cos (b x)-b^2 \cos (a x)}{a^2-b^2} \end{align*}

✓ Sympy. Time used: 0.214 (sec). Leaf size: 104

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

\[ y{\left (x \right )} = \frac {a^{2} e^{x \sqrt {- b^{2}}}}{2 a^{2} - 2 b^{2}} + \frac {a^{2} e^{- x \sqrt {- b^{2}}}}{2 a^{2} - 2 b^{2}} - \frac {b^{2} e^{x \sqrt {- a^{2}}}}{2 a^{2} - 2 b^{2}} - \frac {b^{2} e^{- x \sqrt {- a^{2}}}}{2 a^{2} - 2 b^{2}} \]