75.18.15 problem 604

Internal problem ID [17024]
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 : 604
Date solved : Thursday, March 13, 2025 at 09:10:33 AM
CAS classification : [[_3rd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1\\ y^{\prime \prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 18

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

\[ y = \frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+x^{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 25

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

\[ y(x)\to x^2-\frac {e^{-x}}{2}+\frac {e^x}{2} \]

✓ Sympy. Time used: 0.177 (sec). Leaf size: 17

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

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