39.5.14 problem Problem 24.37

Internal problem ID [6556]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 24. Solutions of linear DE by Laplace transforms. Supplementary Problems. page 248
Problem number : Problem 24.37
Date solved : Wednesday, March 05, 2025 at 12:56:34 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y&=x^{2} {\mathrm e}^{x} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2\\ y^{\prime \prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.167 (sec). Leaf size: 16

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

\[ y = \frac {{\mathrm e}^{x} \left (x^{5}+60 x +60\right )}{60} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

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

\[ y(x)\to \frac {1}{60} e^x \left (x^5+60 x+60\right ) \]

✓ Sympy. Time used: 0.341 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \left (x \left (\frac {x^{4}}{60} + 1\right ) + 1\right ) e^{x} \]