75.18.17 problem 606

Internal problem ID [17034]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 606
Date solved : Monday, March 31, 2025 at 03:38:45 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 25

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

\[ y = -2 x +\frac {4 \,{\mathrm e}^{x}}{3}-\frac {4 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 38

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

\[ y(x)\to \frac {1}{3} \left (-6 x+4 e^x-4 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 31

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

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