71.13.14 problem 14

Internal problem ID [14455]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number : 14
Date solved : Thursday, March 13, 2025 at 03:30:45 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }&=x +\cos \left (x \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 8.973 (sec). Leaf size: 34

ode:=diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)+2*diff(y(x),x) = x+cos(x); 
ic:=y(0) = 1, D(y)(0) = -1, (D@@2)(y)(0) = 2; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = \frac {17 \,{\mathrm e}^{-2 x}}{40}+\frac {x^{2}}{4}-\frac {3 \cos \left (x \right )}{10}+\frac {\sin \left (x \right )}{10}-\frac {3 x}{4}-\frac {{\mathrm e}^{-x}}{2}+\frac {11}{8} \]

✓ Mathematica. Time used: 21.91 (sec). Leaf size: 227

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

\[ y(x)\to \int _1^xe^{-2 K[3]} \left (-\int _1^0-e^{2 K[1]} (\cos (K[1])+K[1])dK[1]+\int _1^{K[3]}-e^{2 K[1]} (\cos (K[1])+K[1])dK[1]-e^{K[3]} \int _1^0e^{K[2]} (\cos (K[2])+K[2])dK[2]+e^{K[3]} \int _1^{K[3]}e^{K[2]} (\cos (K[2])+K[2])dK[2]-1\right )dK[3]-\int _1^0e^{-2 K[3]} \left (-\int _1^0-e^{2 K[1]} (\cos (K[1])+K[1])dK[1]+\int _1^{K[3]}-e^{2 K[1]} (\cos (K[1])+K[1])dK[1]-e^{K[3]} \int _1^0e^{K[2]} (\cos (K[2])+K[2])dK[2]+e^{K[3]} \int _1^{K[3]}e^{K[2]} (\cos (K[2])+K[2])dK[2]-1\right )dK[3]+1 \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = \frac {x^{2}}{4} - \frac {3 x}{4} + \frac {\sin {\left (x \right )}}{10} - \frac {3 \cos {\left (x \right )}}{10} + \frac {11}{8} - \frac {e^{- x}}{2} + \frac {17 e^{- 2 x}}{40} \]