Table of Contents
- 1 Is it possible for two distinct solutions to the equation intersect Why or why not?
- 2 Do differential equations have unique solutions?
- 3 What does the existence and uniqueness theorem tell us about each of the differential equations and initial conditions listed below?
- 4 What is the existence and uniqueness of a solution?
- 5 Does existence and uniqueness theorem guarantee a unique solution to the IVP?
- 6 How do you show a solution is unique?
Is it possible for two distinct solutions to the equation intersect Why or why not?
Two distinct solutions of x′(t)=f(x(t)) cannot intersect at any point.
Do differential equations have unique solutions?
Knowing that a differential equation has a unique solution is sometimes more important than actually having the solution itself! Next, if the interval in the theorem is the largest possible interval on which p(t) and g(t) are continuous then the interval is the interval of validity for the solution.
What does the existence and uniqueness theorem tell us about each of the differential equations and initial conditions listed below?
Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a given initial condition.
Can solutions of a differential equation cross?
4 Solutions Curves Cannot Cross. The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.
Can two solution curves intersect?
If you have 2 intersecting solutions to the same ODE you have 2 solutions to the same IVP (initial value problem) which uniqueness theorems care about (in the case of ODEs). So the trivial answer is: Yes, if they’re equal.
What is the existence and uniqueness of a solution?
The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.
Does existence and uniqueness theorem guarantee a unique solution to the IVP?
However, if we avoid the t-axis—that is, if we choose an initial condition x ( t 0 ) = x 0 ≠ 0 —then the Existence and Uniqueness Theorem guarantees that there will be a unique solution for the IVP.
How do you show a solution is unique?
In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.
Can solution curves cross?
This implies that the isocline must in fact coincide with a solution curve. Hence, one solution to the differential equation (1.3. 4) is y(x) = x + 1, and, by the existence and uniqueness theorem, no other solution curve can intersect this one.
How do you prove that an equation has two solutions?
Between any two roots of f(x), there must be a point of zero derivative, by mean value theorem. If there are more than two roots, then f′(x) would have more than one root, contradiction.