Summary - Calculus and Differential Equations I | MATH 250A, Study notes of Differential Equations

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Calculus and Differential Equations I

MATH 250 A

Summary

What differential equations are and how we study them

Examples of differential equations and of systems of

differential equations.

Identify the independent and dependent variables, as well as the parameters, if any. Characteristic properties: order, linear vs. nonlinear, autonomous vs. non-autonomous. Initial or boundary conditions are often given.

Questions to be addressed in the study of a differential

equation (or a system of differential equations)

Existence and uniqueness Geometric considerations Numerical solutions Analytical solutions

Geometric considerations

Decide where solution curves increase, decrease, or are

concave up or down

For equations of the form y ′^ = g (x) For equations of the form y ′^ = g (y ) You should be able to generalize the above to equations of the form y ′^ = g (x, y )

Symmetries of the family of solution curves S given

symmetries of the differential equation E

If E is invariant under x → −x, then S is symmetric with respect to the y -axis. If E is invariant under y → −y , then S is symmetric with respect to the x-axis. If E is invariant under x → −x and y → −y , then S is symmetric with respect to the origin. Back

Numerical solutions

For equations of the form y ′^ = g (x), we have seen various

ways of approximating integrals.

For equations of the form y ′^ = g (x, y ), we have discussed

Euler’s method.

In both cases, we can sometimes decide whether an

approximation is an underestimate or overestimate.

We also discussed the various approximation errors associated

with these methods.

As part of the above, we introduced Taylor polynomials as

ways to approximate functions and discussed the resulting

error.

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