查看完整版本 : Differential Equations

darigold 2015-12-29 12:57 AM

Differential Equations

[url=http://andrew-exercise.blogspot.com/2015/12/integrating-factor.html]Integrating Factor[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/homogenous-equation.html]Homogenous Equation[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/orthogonal-trajectories.html]Orthogonal Trajectories[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/exact-differential-equation.html]Exact differential equation[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/first-order-linear-differential-equation.html]First order linear differential equation[/url]

Sorry i am not major in Mathematics, but i am interested in it, and i only little about u-level mathematics.

May i know why we need to use the "partial differential"  (i don't know the name of that method) to solve Question one, but not simply applying the ordinary integration method that we learned in M1/M2/A-level?

[[i] 本帖最後由 細路=v= 於 2015-12-29 03:21 PM 編輯 [/i]]

darigold 2015-12-29 11:19 PM

[quote]原帖由 [i]細路=v=[/i] 於 2015-12-29 03:17 PM 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=432886262&ptid=25406487][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]
Sorry i am not major in Mathematics, but i am interested in it, and i only little about u-level mathematics.

May i know why we need to use the "partial differential"  (i don't know the name of that method) to solve Question one, but not simply applying the ordinary integration method that we learned in M1/M2/A-level?[/quote]So am I, not a major in Mathematics, it has been 10 years since I graduate from computer science.
To be honest, I think partial derivative is really just the preferred notation when dealing with multi-variable functions.

Let say, you have a function f(x, y) = x^2 + 2xy + y, what is df/dx?

By definition, we have df/dx = lim (delta -> 0) (f(x + delta) - f(x))/delta, note that in the definition, the function only take one input.

The partial derivative fixed the problem of the definition

pf/px = lim (delta -> 0) (f(x + delta, y) - f(x, y))/delta, see we keep y constant.

The whole result is that we have pf/fx = 2x + 2y, we can also get pf/py = 2x + 1

There isn't anything conceptually different for partial derivative, just keep the other variables as if they're constants!

Another way of looking into it is to think as if we cut the 2d space with a line y = constant, then the function becomes a function of a single variable, and then we just differentiate as usual.

Math aside, I am really happy to meet someone who is interested in mathematics, keep up your learning!

darigold 2015-12-29 11:21 PM

[url=http://andrew-exercise.blogspot.com/2015/12/reduction-of-order-i.html]Reduction of order (I)[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/reduction-of-order-ii.html]Reduction of order (II)[/url]

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darigold 2016-1-1 02:57 PM

[url=http://andrew-exercise.blogspot.com/2015/12/second-order-constant-coefficients-i.html]Second order constant coefficients (I)[/url]

[url=http://andrew-exercise.blogspot.com/2015/12/on-repeated-roots.html]On repeated roots[/url]

darigold 2016-1-2 08:35 AM

[url=http://andrew-exercise.blogspot.com/2016/01/second-order-constant-coefficients-ii.html]Second order constant coefficients (II)[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/second-order-constant-coefficients-iii.html]Second order constant coefficients (III)[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/the-method-of-undetermined-coefficients.html]The method of undetermined coefficients (I)[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/the-method-of-undetermined-coefficients_1.html]The method of undetermined coefficients (II)[/url]

darigold 2016-1-4 08:47 AM

[url=http://andrew-exercise.blogspot.com/2016/01/the-method-of-variation-of-parameter-i.html]The method of variation of parameter (I)[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/the-method-of-variation-of-parameter-ii.html]The method of variation of parameter (II)[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/the-use-of-known-solution-to-find.html]The use of a known solution to find another[/url]

[url=http://andrew-exercise.blogspot.com/2016/01/sturmliouville-theory-show-that-linear.html]Sturm–Liouville theory - show that the linear operator is self-adjoint.[/url]