72.23.4 problem 3 (d)

Internal problem ID [19741]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 50. Applications to differential equations. Problems at page 462
Problem number : 3 (d)
Date solved : Thursday, October 02, 2025 at 04:41:51 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.094 (sec). Leaf size: 23

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

\[ y = 5 \,{\mathrm e}^{-x}+x^{3}-5-3 x^{2}+6 x \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to x^3-3 x^2+6 x+5 e^{-x}-5 \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = x^{3} - 3 x^{2} + 6 x - 5 + 5 e^{- x} \]