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solving initial value problem

Initial Conditions; Initial-Value Problems As we noted in the preceding section, we can obtain a particular solution of an nth order differential equation simply by assigning specific values to the n constants in the general solution. However, in typical applications of differential equations you will be . Free ordinary differential equations (ODE) calculator - solve ordinary differential equations (ODE) step-by-step. InitialValueProblems Aswehaveseen, mostdifferentialequationshavemorethanonesolution. Fora first-orderequation.


Initial value problem - Wikipedia


An initial value problem in the context of a differential equation here, an ordinary differential equation is the following data:. A solution to this is a functional or relational solution to the original differential equation such that atwe have and the derivative is for. More concretely, a function solves the initial value problem if it solves the differential equation andi.

Note that the typical setup of an initial value problem specifies derivatives only up to the for an order differential equation. The intuition is that the differential equation controls the and higher derivatives in terms of the derivatives up to the.

In the case of an explicit differential equationthe and higher derivatives are uniquely determined. In general, they are usually determined up to a finite or discrete solving initial value problem. There are some strategies to solve initial value problems that do not rely on finding the general solution. Generally, these involve doing definite instead of indefinite integrations in the process of finding the solution. This way we avoid introducing unnecessary constants.

In some situations, this can save us from the effort of making cases based on the signs of those unknown constants, because definite integration avoids the introduction of unnecessary parameters.

Instead of a single initial value problem, we may be given the values of the function and some of its derivatives at multiple points. The trade-off is that fewer derivatives may be specified. The solution method is the same: find the general solution, then plug in the conditions solving initial value problem get equations for the free parameters. In general, we would like to have conditions in total values of functions and derivatives in order to get a unique, or finite, solving initial value problem, set of possible solutions, solving initial value problem.

One extreme case of a multiple initial value problem is the case where the value of the function but no derivatives is specified at distinct points of the domain.

Consider the separable differential equation :. This has a stationary solution. The relational solution family is given by:. Let and get NOTE : This has nothing to do with the denoting the order of a differential equation, used elsewhere on this page :.

We can combine back the stationary solution by allowing to get:. We can see from this that every initial value specification yields a unique value of we essentially get a linear equation inand hence every initial value problem for this differential equation has a unique solution.

We can verify that this is both a solution to the differential equation and satisfies the initial value condition. So we get the stationary solution. This is as expected -- for solving initial value problem initial value condition where the initial value of is a stationary solution constant, the stationary solution should be one solution to the initial value problem.

In our case, solving initial value problem, every initial value problem has a unique solution, so it is the only solution. Jump to: navigationsearch. Navigation menu Personal tools Log in. Namespaces Page Discussion. Views Read View source View history. Most viewed methods Integration by parts First derivative test Product rule Second derivative test Piecewise differentiation.

Most viewed general theorems Clairaut's theorem on equality of mixed partials Inverse function theorem Lagrange mean value theorem. Most viewed core concepts Derivative Limit. This page was last edited on 24 Julyat Privacy policy About Calculus Disclaimers Mobile view.

 

 

solving initial value problem

 

Initial Conditions; Initial-Value Problems As we noted in the preceding section, we can obtain a particular solution of an nth order differential equation simply by assigning specific values to the n constants in the general solution. However, in typical applications of differential equations you will be . Free ordinary differential equations (ODE) calculator - solve ordinary differential equations (ODE) step-by-step. InitialValueProblems Aswehaveseen, mostdifferentialequationshavemorethanonesolution. Fora first-orderequation.