you are given an initial condition for the differential equation y\' = y^3 + 3y^

Question

you are given an initial condition for the differential equation y' = y^3 + 3y^2 -10y.

What does the Existence and Uniqueness Theorem tell us about the solution y(t) satisfying the given initial condition?

4. y(0) = 1

5. y(0) = 2

6. y(0) = -1

7. y(0) = -2014.

8. (a) Solve the IVP y' = y^3, y(0) = 1 /2 .

(b) Find the domain of definition of the solution.

(c) Describe what happens to the solution as it approaches the limits of its domain. Why can’t the solution be extended to a larger set of values of t?

9. Consider the dierential equation y' = 5y^4/5.

(a) Show that y(t) = 0, for all t, is an equilibrium solution.

(b) Find a dierent solution satisfying the initial condition y(0) = 0. [Hint: You can use the language like “y(t) = ··· when t 0 and y(t) = ··· when t > 0,

Explanation / Answer

(Existence and Uniqueness Theorem) Given the second order linear equation Let a be any point on the interval I, and let and be any two real numbers.

Then the initial-value problem has y(a) = ; y, (a) =   a unique solution

we cant have multiple values for y(0) unless y has multiple different equations.

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