Finding the differential equation.

[jerelova24]

What is the DE of xsiny=e^(cx)

 

[tkhunny]

I didn't know equations had unique differential equations.

Do you mean, what is the solution to this differential equation? It's separable.

 

[HallsofIvy]

As given, that is not a differential equation so I don't know what tkhunny means by his last line.

We can, of course, differentiate the given expression with respect to x to get
\(\displaystyle \sin(y)+ xcos(y)y'= ce^{cx}\), a differential equation satisfied that expression.

 

[tkhunny]

As given, that is not a differential equation so I don't know what tkhunny means by his last line.

We can, of course, differentiate the given expression with respect to x to get
\(\displaystyle \sin(y)+ xcos(y)y'= ce^{cx}\), a differential equation satisfied that expression.

Well, sorry for abuse. I was just going with the zeroth derivative as it was presented.

 

[Jomo]

xsin(y)+xcos(y)y′=ce^x

Should be sin(y)+xcos(y)y′=ce^cx??

 

[HallsofIvy]

Yes, of course. Thank you. (It's been corrected).

 

[topsquark]

What is the DE of xsiny=e^(cx)

I'm still waiting for the answer to the question: What do you mean by this? There are a large number (in fact, an infinite number) of differential equations you can write where this is the solution.

Is there more to this problem?

-Dan

 

[HallsofIvy]

I was just going with the zeroth derivative as it was presented.

But what do you mean by this? The "zeroth derivative" of a function is just that function. In your original post you had the equation \(\displaystyle x sin(y)= e^{cx}\), an equality of two functions. I still don't understand what you want. Are you asking for an "anti-derivative, a function, f(x,y)= Constant, such that \(\displaystyle \frac{dF(x,y)}{dx}=x sin(y)- e^{cx}\).

 

[tkhunny]

But what do you mean by this? The "zeroth derivative" of a function is just that function. In your original post you had the equation \(\displaystyle x sin(y)= e^{cx}\), an equality of two functions. I still don't understand what you want. Are you asking for an "anti-derivative, a function, f(x,y)= Constant, such that \(\displaystyle \frac{dF(x,y)}{dx}=x sin(y)- e^{cx}\).

Over-thinking. It was an ill-formed question and I provided an similarly-motivated response. No rigor intended.

 

[HallsofIvy]

I agre{e that this was an ill formed question. I would assume that the OP meant the simplest DE that has the given function as the general solution. I previously wrote "\(\displaystyle sin(y)+ x cos(y)\frac{dy}{dx}= ce^{cx}\)". Of course, that is not the answer because it still has "c" in it.

We can then use the original equation \(\displaystyle xsin(y)= e^{cy}\) to write \(\displaystyle sin(y)+ xcos(y)\frac{dy}{dx}= c x sin(y)\). Finally from the original equation, \(\displaystyle ln(x sin(y))= cy\) so \(\displaystyle c= \frac{ln(x sin(y)}{y}\) and then \(\displaystyle sin(y)+ xcos(y)\frac{dy}{dx}= \frac{ ln(x sin(y)) x sin(y)}{y}\).