Number of initial conditions required to determine unique solutions to ODEs

Question

I am enrolled in an introductory course in differential equations and I recently read in a book that the number of initial conditions required to determine unique solutions to a diff eqn of order 'n' is n. The reason was stated to be that "roughly" there are n-integrations which produce n constants that need to be specified/determined. Can anyone make this intuitive explanation more rigorous/clear?

Note: I apologize for the vague sounding question, but I have no idea what a rigorous explanation appears like so I can't make my question more clear either.

Answer 1

Let us look at a very simple example. $$ y''+3y'-4 =0$$

In order to solve it we look at the characteristic equation, $$p(\lambda)=\lambda ^2 +3\lambda-4 =0$$

Which gives us $ \lambda =1$ and $ \lambda =-4$

You get two linearly independent solutions, $y=e^t$ and $y=e^{-4t}$

The general solution is $$ y=c_1 e^t + c_2 e^{-4t}$$ where $c_1$ and $c_2$ are arbitrary constants to be found from the given initial information.

Similar calculation solves linear higher ordered equations with constant coefficients.

For more complicated equations they use ideas from linear algebra to show the solution set is a vector space with the same dimension as the order of the equation.

Answer 2

A general solution to a $1$st order differential equation will contain a constant $c$. The initial condition can be substituted in to solve for $c$ and hence provide a specific solution. When the order is $n=2$, this involves a double integration and hence solving for two constants which require two initial conditions. In simpler practical terms, A double integration of acceleration, say $\frac{d^2s}{dt^2} = a(t) = 32$, would first be, $v(t) = 32t + c$, and secondly $s(t) = 16t^2 + ct + b$. Without the two initial conditions of velocity and displacement we are unable to provide a specific solution. Hence $n$ initial conditions are needed for a unique solution to differential equations of order $n$.

Answer 3

Any equation of the order $n$ is equivalent to a system of $n$ equations of the first order $$ \dot x_1=f_1(t,x_1,\ldots,x_n),\\ \ldots\\ \dot x_n=f_n(t,x_1,\ldots,x_n). $$ Geometrically it means that at each point $(t,x_1,\ldots,x_n)$ you have a direction along which your solution is pointed. Hence geometrically solve your ode is tantamount to finding a curve that starts at a given point $(t_0,x_1^0,\ldots,x_n^0)$ and has specific directions. Hence in general you will need the initial time noment and $n$ coordinates as your initial conditions.