88.22.5 problem 9

Internal problem ID [24165]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 160 (Laplace transform)
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:00:22 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+4 y^{\prime \prime }&=16 \,{\mathrm e}^{2 x} \end{align*}

Using Laplace method

✓ Maple. Time used: 0.056 (sec). Leaf size: 63

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-4*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x) = 16*exp(2*x); 
dsolve(ode,y(x),method='laplace');
 

\[ y = -3+y \left (0\right )+\frac {{\mathrm e}^{2 x} \left (x y^{\prime \prime \prime }\left (0\right )-2 x y^{\prime \prime }\left (0\right )+8 x^{2}-y^{\prime \prime \prime }\left (0\right )+3 y^{\prime \prime }\left (0\right )-16 x +12\right )}{4}-\frac {y^{\prime \prime }\left (0\right ) \left (3+4 x \right )}{4}+\frac {y^{\prime \prime \prime }\left (0\right ) \left (1+x \right )}{4}+x \left (-2+y^{\prime }\left (0\right )\right ) \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 40

ode=D[y[x],{x,4}]-4*D[y[x],{x,3}]+4*D[y[x],{x,2}]==16*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^{2 x} \left (8 x^2+(-16+c_2) x+12+c_1-c_2\right )+c_4 x+c_3 \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*exp(2*x) + 4*Derivative(y(x), (x, 2)) - 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{4} e^{2 x} + x \left (C_{2} + \left (C_{3} + 2 x\right ) e^{2 x}\right ) \]