1. ## Implicit Solutions

I have two problems that I'm having some serious difficulty with. It's probably because of my lack of knowledge of implicit solutions. If, while you solve these, you could explain a little more about implicit solutions I would be incredibly grateful.

In these problems, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicity as a function of x and use implicit differentiation.

1.
$\displaystyle e^{xy} + y = x - 1$,

$\displaystyle \frac{dy}{dx} = \frac{e^{-xy} - y}{e^{-xy} + x}$

2.
$\displaystyle \sin{(y)} + xy - x^3 = 2$,

$\displaystyle y'' = \frac{6xy' + (y')^3\sin{(y)} - 2(y')^2}{3x^2 - y}$

2. Originally Posted by Aryth
I have two problems that I'm having some serious difficulty with. It's probably because of my lack of knowledge of implicit solutions. If, while you solve these, you could explain a little more about implicit solutions I would be incredibly grateful.

In these problems, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicity as a function of x and use implicit differentiation.

1.
$\displaystyle e^{xy} + y = x - 1$,

$\displaystyle \frac{dy}{dx} = \frac{e^{-xy} - y}{e^{-xy} + x}$
Implicit differentiation is just differentiating an arbitrary equation, instead of differentiating y=...

What you can do in order to solve this problem is first differentiate the equation and then substitute $\displaystyle \frac{dy}{dx}$ you're given and see if the equation is satisfied.

$\displaystyle e^{xy}+y=x-1$
Remember that y is a function of x.
So the derivative of y wrt x will be $\displaystyle \frac{dy}{dx}$ and not 0.
In these problems, you'll mainly have to use the product rule, chain rule and quotient rule.

Let's do it by steps.
What if we differentiate $\displaystyle e^{xy}$ with respect to x ?
We know that its derivative is $\displaystyle (xy)' e^{xy}$
Now what is the derivative of xy : (xy)' ?
Remember that y is a function of x. It's not a constant.
So you have to use the chain rule :
$\displaystyle (xy)'=(x)'y+x(y)'=1 \cdot y+x \cdot \frac{dy}{dx}=y+x \cdot \frac{dy}{dx}$
Hence differentiating $\displaystyle e^{xy}$ will give :
$\displaystyle \left(y+x \cdot \frac{dy}{dx}\right) e^{xy}$

What if we differentiate y with respect to x ?
That's simply $\displaystyle \frac{dy}{dx}$

What if we differentiate x+1 with respect to x ?
That's just 1.

So if you differentiate your equation, you'll get :
$\displaystyle \left(y+x \cdot \frac{dy}{dx}\right) e^{xy}+\frac{dy}{dx}=1$

now substitute $\displaystyle \frac{dy}{dx}$ by the formula you are given and see if the equation is satisfied. And you'll be done.

3. Awesome, got it. The problem I had was manipulating it to look exactly like the differential equation. Thanks for the help.