Prove outputs of diff equations, using formulas of derivatives

oleksiy

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Jun 23, 2019
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9
Hi everybody, I am trying to solve two equations.

y'[x] = (r y[x])/x^2 with y[1] = a
which has a solution a*E^(r - r/x)
and
y'[x] = Sin[x] y[x] with y[0] = a
which has a solution a*E^(1 - Cos[x])

For the first one I wrote something like
We have
y'[x] = (r y[x])/x^2
Hence, y[x] = e^(-r/x) + c
where c is constant
Our y[1] = a
We are given in the output that
a*E^(r - (r/x))

Then the derivative of our output is
are^(r-(r/x))/x^2
In turn
are^(r-(r/x))/x^2


Then I stumbled

For the second one, I wrote similar logic ans stumbled too.

Can somebody please give a tip on how to approach it?

Thanks!
 

Dr.Peterson

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Nov 12, 2017
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5,056
y'[x] = (r y[x])/x^2 with y[1] = a
which has a solution a*E^(r - r/x)
...
For the first one I wrote something like
We have
y'[x] = (r y[x])/x^2
Hence, y[x] = e^(-r/x) + c
where c is constant
Our y[1] = a
We are given in the output that
a*E^(r - (r/x))

Then the derivative of our output is
are^(r-(r/x))/x^2
In turn
are^(r-(r/x))/x^2
You stumbled right at the beginning. It is not true that y(x) = e^(-r/x) + c.

Please show the steps you took to get to that point.

Then, I'm not sure what you are trying to do after that. If your goal is to find c, you don't need to differentiate anything to find it; you need to use the initial condition, which is a condition on y, not y'. Or are you just trying to demonstrate that the solution provided is valid, without having obtained it for yourself?
 

oleksiy

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Jun 23, 2019
Messages
9
Thanks. I am given a problem and an output. Need to use the derivative formulas to explain the output.
 

Subhotosh Khan

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Thanks. I am given a problem and an output. Need to use the derivative formulas to explain the output.
Please post the exact problem given to you - verbatim.

Then include your work/thought.
 

oleksiy

New member
Joined
Jun 23, 2019
Messages
9
Thanks, it looks like this

Here's the formula for a solution of the differential equation
y'[x] = (r y[x])/x^2 with y[1] = a
The solution is a*E^(r - r/x)
Use the derivative formulas you know to explain the output.

I am still working on this and the second one though.
 

Dr.Peterson

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Nov 12, 2017
Messages
5,056
Are you saying that you are not expected to solve the differential equation, only to verify that the given solution is valid? (Terms like "explain" and "output" are confusing.)

What did you find to be the derivative of the given solution?
 

HallsofIvy

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Jan 27, 2012
Messages
5,286
If I were asked to solve the equation \(\displaystyle y'= ry(x)/x^2\) the first thing I would do is write it \(\displaystyle \frac{dy}{dx}= \frac{ry}{x^2}\) and then "separate variables": \(\displaystyle \frac{dy}{y}= r\frac{dx}{x^2}\). Now integrate both sides.

If the problem is, rather, to prove that \(\displaystyle y(x)= ae^{x- r/x}\) then differentiate to get \(\displaystyle y'= a(1+ r/x^2)e^{x- r/x}\) and y into the equation. Is \(\displaystyle a(1+ r/x^2)e^{x- r/x}= \frac{rae^{x- r/x}}{x^2}\) true?

By the way, standard function notation is "y(x)", not "y[x]" and the exponential is "\(\displaystyle e^x\)", not "\(\displaystyle E^x\)".[/tex][/tex]
 
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oleksiy

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Jun 23, 2019
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Are you saying that you are not expected to solve the differential equation, only to verify that the given solution is valid? (Terms like "explain" and "output" are confusing.)

What did you find to be the derivative of the given solution?
Well, I fully quoted the task :(. Sorry for confusion :(
Thanks! If I understand your question correctly, the derivative of the output is (a(x-1)e^(r-r/x))/x)
 
Last edited:

oleksiy

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Jun 23, 2019
Messages
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By the way, standard function notation is "y(x)", not "y[x]" and the exponential is "\(\displaystyle e^x\)", not "\(\displaystyle E^x\)".[/tex][/tex]
Thanks, I just copied the code from Mathematica.
 
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