75.18.17 problem 606

Internal problem ID [17105]
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 : Tuesday, January 28, 2025 at 09:52:49 AM
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*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 25

dsolve([diff(y(x),x$3)-y(x)=2*x,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 2],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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