7.11.35 problem 36

Internal problem ID [356]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 36
Date solved : Saturday, March 29, 2025 at 04:51:33 PM
CAS classification : [[_high_order, _missing_y]]

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

With initial conditions

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 31

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-4*diff(diff(y(x),x),x) = x^2; 
ic:=y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = -1, (D@@3)(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {x^{2}}{16}-\frac {x^{4}}{48}-\frac {3 \,{\mathrm e}^{-2 x}}{64}-\frac {11 \,{\mathrm e}^{2 x}}{64}+\frac {5 x}{4}+\frac {7}{32} \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 44

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

\[ y(x)\to \frac {1}{192} e^{-2 x} \left (e^{2 x} \left (-4 x^4-12 x^2+240 x+42\right )-33 e^{4 x}-9\right ) \]

✓ Sympy. Time used: 0.192 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = - \frac {x^{4}}{48} - \frac {x^{2}}{16} + \frac {5 x}{4} - \frac {11 e^{2 x}}{64} + \frac {7}{32} - \frac {3 e^{- 2 x}}{64} \]