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Math Help - Need explanation of differential equations

  1. #1
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    Need explanation of differential equations

    Hello, I am a student at the University of Pittsburgh. I am majoring in electrical engineering. I have already successfully completed Calculus 1 and 2. Now I have a class in Differential Equations. I have heard that electrical engineers use differential equations a lot, so you see how important it is that I understand them.

    So my situation is this. Yesterday was my first day of Differential Equations class, and as soon as my professor began explaining I knew I was in trouble. To be honest I have no clue what my professor was talking about.

    After I got home I tried reading my book and I still don't understand it. Tomorrow is my second class, and I want to figure this out BEFORE I come to it.

    Okay so I know what a function is. Example: f(x) = x+5

    This means that you input values of X and get outputs of Y.

    I know what a derivative of the above function is. Its a function of the instantaneous rate of change or slope. And the derivative of the above is 2X.

    So just what in heck is a differential equation? Can you please explain using the example above if possible? Please use simple examples because if I can understand the key concept I can do more complex stuff.

    Its the key concept I'm having trouble understanding.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by blackbird View Post
    Hello, I am a student at the University of Pittsburgh. I am majoring in electrical engineering. I have already successfully completed Calculus 1 and 2. Now I have a class in Differential Equations. I have heard that electrical engineers use differential equations a lot, so you see how important it is that I understand them.

    So my situation is this. Yesterday was my first day of Differential Equations class, and as soon as my professor began explaining I knew I was in trouble. To be honest I have no clue what my professor was talking about.

    After I got home I tried reading my book and I still don't understand it. Tomorrow is my second class, and I want to figure this out BEFORE I come to it.

    Okay so I know what a function is. Example: f(x) = x+5

    This means that you input values of X and get outputs of Y.

    I know what a derivative of the above function is. Its a function of the instantaneous rate of change or slope. And the derivative of the above is 2X.

    So just what in heck is a differential equation? Can you please explain using the example above if possible? Please use simple examples because if I can understand the key concept I can do more complex stuff.

    Its the key concept I'm having trouble understanding.
    Check this out. Try to read and understand the first post. If its a little too advanced, let me know. I'll try to simplify it further.

    --Chris
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  3. #3
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    Quote Originally Posted by blackbird View Post
    So just what in heck is a differential equation? Can you please explain using the example above if possible? Please use simple examples because if I can understand the key concept I can do more complex stuff.
    In high school you learn algebraic equations. That is an equation where you have to find a number that satisfies it. For example, x^2 - 2x + 1 =0. You need to find a number which makes this satement true.

    But there is something called a functional equation or in your case a differencial equation. You are not solving for a number. You are solving for a function which makes the equation true. For example, y ' = x. You are solving for y. You want to find a function y such that when you take its derivative you get x. What is the solution? Well, that is just the anti-derivative of x i.e. what function has derivative x. This is \tfrac{1}{2}x^2. That is one example of such a function. But so if \tfrac{1}{2}x^2 + 1. And so is \tfrac{1}{2}x^2-1. In fact the term in the end does not matter. It can be any number because its derivative is zero so it goes away. The full solution to the equation y' = x is therefore \tfrac{1}{2}x^2 + k where k is any number. (Note if you find \smallint x dx you will get precisely this result).

    Unlike algebraic equations which has one solution of just a few solutions, differencial equations a lot of times have infinitely many solutions. Like the example above. This above example is really simple. Differencial equations get more advanced. For example, y ' = y. This means find a function whose derivative is equal to itself. You ought to know e^x is one such function. But that is not the whole solution. Note that 2e^x works too. In fact, the whole solution to this equation is ke^x where k is an arbitrary number. Thus, there are again infinitely many solutions.

    What differencial equations are about is solving this types of equations. How to find solutions and all that stuff.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by blackbird View Post
    Hello, I am a student at the University of Pittsburgh. I am majoring in electrical engineering. I have already successfully completed Calculus 1 and 2. Now I have a class in Differential Equations. I have heard that electrical engineers use differential equations a lot, so you see how important it is that I understand them.

    So my situation is this. Yesterday was my first day of Differential Equations class, and as soon as my professor began explaining I knew I was in trouble. To be honest I have no clue what my professor was talking about.

    After I got home I tried reading my book and I still don't understand it. Tomorrow is my second class, and I want to figure this out BEFORE I come to it.

    Okay so I know what a function is. Example: f(x) = x+5

    This means that you input values of X and get outputs of Y.

    I know what a derivative of the above function is. Its a function of the instantaneous rate of change or slope. And the derivative of the above is 2X.

    So just what in heck is a differential equation? Can you please explain using the example above if possible? Please use simple examples because if I can understand the key concept I can do more complex stuff.

    Its the key concept I'm having trouble understanding.
    differential equations can get quite involved, but as for what they are, it is simple, in terms of its definition at least

    a differential equation is simply an equation in which the unknowns are actually a function and some of its derivatives, and you are required to find the function.

    example, y' - y = 0 is a differential equation. y' and y are the unknowns. recall that y' means the derivative of y with respect to some variable. usually x, but the common variable in diff eq's is t, so y' = \frac {dy}{dt} usually, that is, the derivative of y with respect to t. now, to solve a differential equation, you need to find what the function y is. as in, find the class or family of functions for which this equation holds true. if more information is given, you can find the particular function that works. for instance, one particular function that works here is y = e^x. since (e^x)' - e^x = e^x - e^x = 0

    there are various methods to solve differential equations, as there are many kinds. you will be learning about them in your class. good luck, hope that helped


    EDIT: geez, two times too late. i got the bronze medal here
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Jhevon View Post
    EDIT: geez, two times too late. i got the bronze medal here
    Haha!

    Again, I take the gold

    --Chris
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  6. #6
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    Thank you very much, I think I understand this now. I gotta admit its a little bit weird compared to everything I have been doing for the past year in Calculus. Well its kinda sorta like implicit differentiation I guess.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by blackbird View Post
    Thank you very much, I think I understand this now. I gotta admit its a little bit weird compared to everything I have been doing for the past year in Calculus. Well its kinda sorta like implicit differentiation I guess.
    it gets easier. i was lost for the first few days in diff eqs also
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