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<h3>
Answer: 81</h3>
Work Shown:
x = number of cats
(4 cats)/(5 dogs) = (x cats)/(45 dogs)
4/5 = x/45
4*45 = 5x
180 = 5x
5x = 180
x = 180/5
x = 36
There are 36 cats and x+45 = 36+45 = 81 cats and dogs total.
Answer:
The functions satisfy the differential equation and linearly independent since W(x)≠0
Therefore the general solution is

Step-by-step explanation:
Given equation is

This Euler Cauchy type differential equation.
So, we can let

Differentiate with respect to x

Again differentiate with respect to x

Putting the value of y, y' and y'' in the differential equation



⇒m²-10m +24=0
⇒m²-6m -4m+24=0
⇒m(m-6)-4(m-6)=0
⇒(m-6)(m-4)=0
⇒m = 6,4
Therefore the auxiliary equation has two distinct and unequal root.
The general solution of this equation is

and

First we compute the Wronskian


=x⁴×6x⁵- x⁶×4x³
=6x⁹-4x⁹
=2x⁹
≠0
The functions satisfy the differential equation and linearly independent since W(x)≠0
Therefore the general solution is
