78.2.44 problem 18

Internal problem ID [20996]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 18
Date solved : Thursday, October 02, 2025 at 07:01:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }-15 y&=x^{4} {\mathrm e}^{x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 64

ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)-15*y(x) = x^4*exp(x); 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {3+8 \left (x^{7}-7 x^{6}+42 x^{5}-210 x^{4}+840 x^{3}-2520 x^{2}+5040 x -5040\right ) {\mathrm e}^{x}-x^{8} {\mathrm e}+5 x^{8}+14833 \,{\mathrm e}}{8 x^{5}} \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 65

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]-15*y[x]==x^4*Exp[x]; 
ic={y[1]==1,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {5 x^8-e \left (x^8-14833\right )+8 e^x \left (x^7-7 x^6+42 x^5-210 x^4+840 x^3-2520 x^2+5040 x-5040\right )+3}{8 x^5} \end{align*}

✓ Sympy. Time used: 0.496 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4*exp(x) + x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) - 15*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\frac {x^{8} e^{x}}{8} + x^{8} \left (\frac {5}{8} - \frac {e}{8}\right ) + \frac {\left (- x^{8} + 8 x^{7} - 56 x^{6} + 336 x^{5} - 1680 x^{4} + 6720 x^{3} - 20160 x^{2} + 40320 x - 40320\right ) e^{x}}{8} + \frac {3}{8} + \frac {14833 e}{8}}{x^{5}} \]