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This tag is about questions regarding Initial value problems. In the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.

A lot of the equations that we deal with in science and engineering are derived from a specific type of differential equation called an initial value problem.

An Initial Value Problem $($often times abbreviated I.V.P. and also called a Cauchy problem$)$ is a problem where we want to find a solution to some differential equation that satisfies an appropriate number of initial conditions.

Generalized Definition: An Initial Value Problem for an $~n^{\text{th}}~$ order ordinary differential equation is an $~n^{\text{th}}~$ order ODE $$y^{(n)}=h(t,y,y^{(1)},\cdots,y^{(n−1)})$$ with initial Conditions $~y^{(i−1)}(τ)=ξ_i~$ where $~i=1,2,\cdots,n~$ and $~(τ,ξ_1,ξ_2,\cdots,ξ_n)~∈~D~$.

A Solution to the initial value problem $$y^{(n)}=h(t,y,y^{(1)},\cdots,y^{(n−1)})$$ with initial conditions $~y^{(i−1)}(τ)=ξ_i~$ where $~i=1,2,\cdots,n~$ on the open interval $~J=(a,b)~$ is an $n-$times continuously differentiable function $~ϕ∈C^n(J,\mathbb R)~$ such that for all $~t∈J~$ we have that $~(t,ϕ(t),ϕ^{(1)}(t),\cdots,ϕ^{(n−1)}(t))∈D~$, $~ϕ(n)=h(t,ϕ,ϕ^{(1)},\cdots,ϕ^{(n−1)})~$, and $~ϕ^{(i−1)}(τ)=ξ_i~$ for $~i=1,2,\cdots,n~$.

Differences between Initial and Boundary Value Problems :

An initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term “initial” value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable.

Note: In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential initial value is an equation that is an evolution equation specifying how, given initial conditions, the system will evolve with time.

References:

https://en.wikipedia.org/wiki/Initial_value_problem

"Differential Equations" by Shepley L. Ross

"Differential Equations with Applications and Historical Notes " by George Simmons

"Differential Equations: Theory, Technique, and Practice" by George F. Simmons and Steven G. Krantz