12.9.5 problem 5

Internal problem ID [1761]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 5
Date solved : Saturday, March 29, 2025 at 11:38:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=7 x^{{3}/{2}} {\mathrm e}^{x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = 7*x^(3/2)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (c_2 +c_1 x +\frac {4 x^{{7}/{2}}}{5}\right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 29

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==7*x^(3/2)*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{5} e^x \left (4 x^{7/2}+5 c_2 x+5 c_1\right ) \]

✓ Sympy. Time used: 0.244 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-7*x**(3/2)*exp(x) + y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {4 x^{\frac {5}{2}}}{5}\right )\right ) e^{x} \]