How many solutions does this equation have?
2 answers:
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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
Hi!
I would approach this question by first looking at where the parallelogram is located, by graphing the image. I did so and drew a rough image on Paint (See attached image)
To get the part where the diagonals intersect, I would find the midpoint of the line between points (1, 3) and (5,-9) (or the other pair). The reason is a parallelogram's diagonals always bisect each other, meaning the point they intersect is always the middle of the two diagonals.
Therefore, you can find the midpoint of a diagonal, between (1, 3) and (5, -9). The midpoint theorem is (.
Take the points (1, 3) and (5, -9), and fill them in.
Then solve.
(3, -3)
If you'd like to check the other midpoint:
Take the points, (8, 3) and (-2, -9)
Then solve.
(3, -3)
They're the same, so that answer is correct.
Hope this helps!
