# 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.

• topsquark

#### 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.

#### Jomo

##### Elite Member
• topsquark

#### HallsofIvy

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

• Jomo

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

• topsquark