# Thread: [SOLVED] Challenging differential equation

1. ## [SOLVED] Challenging differential equation

I would really appreciate some assistance with the following problem. I barely know where to start!

Let x+y=z

Solve the equations a) and b):

a) d/dx [ (1/3)x^3+x+2 ] = 0, where x is an element of the set C
Solve for x.

b) dy/du = ln(e), y(1)=1
Define the function y.

2. Originally Posted by aujt74 Let x+y=z

a) d/dx [ (1/3)x^3+x+2 ] = 0, where x is an element of the set C
Solve for x.

Here's a kick start.
$\displaystyle \frac{d}{dx}\left[\frac{1}{3}x^3+x+2\right] = x^2+1$

Now solve

$\displaystyle x^2+1 = 0$ Originally Posted by aujt74 Let x+y=z

b) dy/du = ln(e), y(1)=1
Define the function y.
also consider that $\displaystyle \ln(e)=1$

3. Thanks for that kickstart!

$\displaystyle x^2+1 = 0$
so $\displaystyle x= i$

$\displaystyle \frac{dy}{du}=\ln(e) = 1$
so integrating wrt to u $\displaystyle y = u$ ?

That's right isn't it?

4. Originally Posted by aujt74 $\displaystyle \frac{dy}{du}=\ln(e) = 1$
so integrating wrt to u $\displaystyle y = u$ ?

That's right isn't it?
Yes. In general, $\displaystyle y = u + C$, but you have $\displaystyle y(1) = 1$, so $\displaystyle 1 = 1 + C$ and $\displaystyle C = 0$.

5. Originally Posted by aujt74 $\displaystyle x^2+1 = 0$
so $\displaystyle x= i$
$\displaystyle x = \pm i$

6. Thanks for all the help so far! Originally Posted by aujt74 Let x+y=z
$\displaystyle x = \pm i$

$\displaystyle y = u$

so $\displaystyle z =$ ?

I don't understand how to evaluate $\displaystyle z$? I've not been given any values for u.

7. $\displaystyle z= \pm i +u$

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