Finding the differential equation.

jerelova24

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

tkhunny

Moderator
Staff member
I didn't know equations had unique differential equations.

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

HallsofIvy

Elite Member
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

Moderator
Staff member
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.

HallsofIvy

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

topsquark

Full Member
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

Elite Member
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

Moderator
Staff member
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

Elite Member
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 $$\displaystyle sin(y)+ xcos(y)\frac{dy}{dx}= \frac{ ln(x sin(y)) x sin(y)}{y}$$.