Answer:
y = 4
Step-by-step explanation:
The form of the exponential function is y = a
Use the given points to solve for a and b
Using (0, 4 ), then
4 = a ( = 1 ), thus a = 4
y = 4
Using (2, 196 ), then
196 = 4b² ( divide both sides by 4 )
49 = b² ( take the square root of both sides )
b = = 7
y = 4 ← exponential function
We want a solution in the form
with derivatives
Substituting and its derivatives into the ODE,
gives
Shift the index on the second sum to have it start at :
and take the first term out of the other two sums. Then we can consolidate the sums into one that starts at :
and so the coefficients in the series solution are given by the recurrence,
or more simply, for ,
Note the dependency between every other coefficient. Consider the two cases,
- If , where is an integer, then
and so on, with the general pattern
- If , then
and we would see that for all .
So we have
so that one solution is
and the other is
I've attached a plot of the exact and series solutions below with , , and to demonstrate that the series solution converges to the exact one.
Answer:
u = x²
Step-by-step explanation:
4x⁴ − 21x² + 20 = 0
Substitute u = x² to turn the equation into a quadratic.
4u² − 21u + 20 = 0
You can solve by factoring.
(4u − 5) (u − 4) = 0
u = 5/4 or 4
x² = 5/4 or 4
x = ±½√5 or ±2
Well, solving for X it would be X = -62/2