PDEs 4: Linear ODEs with Constant Coefficients, Finding a Particular Solution
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- Опубликовано: 18 сен 2024
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Professor MathTheBeautiful, thank you for a short and sweet video/lecture on Linear ODEs with Constant Coefficients. This is an error free video/lecture on RUclips.
So glad I found this lecture series. Its excellent! ❤ Great job!
Thank you, glad you're enjoying it!
Great video! We just learned this in class but from a purely systematic calculus perspective. It's really interesting to see an approach using concepts from linear algebra.
Thank you sm
How about equations for which the exponential function won't work? E.g. this one:
x²·d²y/dx² + 3·x·dy/dx = 15·y
One possible solution is `y = x³`, which is not an exponential.
First note, for the above method to apply the equation sbould have constant coefficients. x^2 is not constant coefficient. Neither is 3x.
Second note, we can solve the general solutions to it by method called Reduction of Order. which requires that we know one of the solution p(t)y'' + q(t) y' + r(t) = 0, exactly in your sceniaro.
I know that the coefficients aren't constant. That's exactly why I am asking about this.
I'm looking for the general method of solving such equations even if the coefficients are not constant, but some polynomial functions of the independent variable.
I know about the reduction of order technique, but it applies only to a small subclass of equations when one of the terms is missing, and - as you pointed out - only when we know at least one of the solutions.
Sure, we can always use power series methods to solve any equation, but once we go to that territory, there is usually no going back, because - as far as I know - there are no known methods to convert a power series back to the original function when the series doesn't look like any of the series expansions of the functions we're familiar with. It's like solving an equation for a number and getting some formula for the digits of its decimal expansion, but not being able to convert it back to a real number.
I see this lack of methods as quite limiting, because 2nd order equations with non-constant coefficients appear quite often in real-life problems in physics.
this specific one would be an Euler equation
y^(n)x^some different different degree than n+y^(n-1)x^again, some kind of degree+....+yx=inhomogeneous side has no general solution
anybody tell me what EIGENVALUE Function is ????
It's a scaler value associated with a linear transformation , Please see 3blue1brown for more details
Very good teacher, but still... in this series I learned nothing about PDEs, except that they are difficult to solve
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