### Differential equations: Direction field

### Solution strategy on the basis of the slope field

**Asymptotics of an initial value problem** is concerned with the behavior of solutions when the time \(t\) approaches a boundary of the maximum existence interval. A solution can simply "explode", but it can also approach a specific solution. In either case, the direction field can help to find a solution.

General solution strategy for a first-order ODE of first degree

The following approach to an ODE of the form #y'=\varphi(t,y)# can help in finding its general solution.

- Draw the direction field
- Check the direction field to see if there are special solutions, such as a periodic, an equilibrium solution, or points where a specific solution meets a boundary of the maximum existence interval.
- Set up a conjectural form of that solution in terms of known functions with some parameters
- Substitute this conjectural solution into the differential equation and see if this leads to equations in the parameters that have a solution; if so, then a solution has been found.

- Check the behavior of solutions near the solution curve associated with the solution found, and replace #s(t)# with a suitable adjustment #s(t)# by a general function #u(t)#.
- Transform the equation in #y# into a differential equation in #u# (which is simpler than the original ODE) and solve it.

There is no guarantee that this strategy works. *Later* we will get to know special cases in which solutions can always be found.

We illustrate this approach by means of some examples.

Consider the following differential equation: \[\frac{\dd y}{\dd t}=t-y\] The direction field of this differential equation is shown in the figure below, along with some solution curves. In this example, each solution curve converges to the black solution curve. Determine this specific solution #s(t)#.

Next use the specific solution #s(t)# to find the general solution in the form #y(t)=u(t)+s(t)# for a function #u(t)# that is still to be determined.

*later*be able to recognize this type of differential equation as linear of first order. There is a general solution methode for this type of ODE.

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