20.11.10 problem Problem 13

Internal problem ID [3792]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 13
Date solved : Sunday, March 30, 2025 at 02:08:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=15 \,{\mathrm e}^{3 x} \sqrt {x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+9*y(x) = 15*exp(3*x)*x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} \left (4 x^{{5}/{2}}+c_1 x +c_2 \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 25

ode=D[y[x],{x,2}]-6*D[y[x],x]+9*y[x]==15*Exp[3*x]*Sqrt[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{3 x} \left (4 x^{5/2}+c_2 x+c_1\right ) \]

✓ Sympy. Time used: 0.231 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-15*sqrt(x)*exp(3*x) + 9*y(x) - 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + 4 x^{\frac {3}{2}}\right )\right ) e^{3 x} \]