88.22.7 problem 11

Internal problem ID [24167]
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 : 11
Date solved : Thursday, October 02, 2025 at 10:00:23 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

Using Laplace method

✓ Maple. Time used: 0.137 (sec). Leaf size: 120

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

\[ y = -\frac {53}{4}-2 x^{3}-\frac {53 x}{2}-\frac {5 x^{2}}{2}-x^{4}+\frac {{\mathrm e}^{x} \left (2 y \left (0\right )+y^{\prime }\left (0\right )+y^{\prime \prime }\left (0\right )-y^{\prime \prime \prime }\left (0\right )+46\right )}{3}+\frac {\left (-4 y \left (0\right )+4 y^{\prime \prime \prime }\left (0\right )-5\right ) {\mathrm e}^{2 x}}{28}+\frac {\left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \left (-40+10 y \left (0\right )-7 y^{\prime }\left (0\right )-7 y^{\prime \prime }\left (0\right )+4 y^{\prime \prime \prime }\left (0\right )\right )+\sqrt {3}\, \left (-2 y \left (0\right )+2 y^{\prime \prime \prime }\left (0\right )-7 y^{\prime \prime }\left (0\right )+7 y^{\prime }\left (0\right )+148\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{21} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 87

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

\begin{align*} y(x)&\to -x^4-2 x^3-\frac {5 x^2}{2}-\frac {53 x}{2}+c_3 e^x+c_4 e^{2 x}+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )-\frac {53}{4} \end{align*}

✓ Sympy. Time used: 0.225 (sec). Leaf size: 66

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

\[ y{\left (x \right )} = C_{3} e^{x} + C_{4} e^{2 x} - x^{4} - 2 x^{3} - \frac {5 x^{2}}{2} - \frac {53 x}{2} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} - \frac {53}{4} \]