78.24.2 problem 3 (b)

Internal problem ID [18388]
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 51. Derivatives and Integrals of Laplace Transforms. Problems at page 467
Problem number : 3 (b)
Date solved : Monday, March 31, 2025 at 05:27:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.140 (sec). Leaf size: 37

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

\[ y = \frac {-\delta \left (x \right )+x \,{\mathrm e}^{-x} \left (\delta \left (2, 0\right )+\delta \left (1, 0\right )\right )-\delta \left (1, x\right )}{\delta \left (2, 0\right )+\delta \left (1, 0\right )} \]

✗ Mathematica

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

{}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*y(x)*exp(x) - x*exp(x)*Derivative(y(x), (x, 2)) - 3*y(x)*exp(x) + 3)*exp(-x)/(2*x + 3) cannot be solved by the factorable group method