# General solution of differential equation (express y in term of x)

1. May 3, 2014

### delsoo

1. The problem statement, all variables and given/known data

i got stucked here. below is the answer given. can anybody help please?

2. Relevant equations

3. The attempt at a solution

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2. May 3, 2014

### haruspex

You can try to simplify the original equation by substituting y(x) = f(x).xn, then see what value of n will get rid of the -y/x term.

3. May 3, 2014

### Staff: Mentor

This looks like a straightforward "integrating factor" problem.

Chet

4. May 3, 2014

### delsoo

dy/dx +Py(x) = Q(X) if i rearrange i would get 0.5 dy/dx + y/x = arc tan x ... is arc tan x function of x?

Last edited: May 3, 2014
5. May 3, 2014

### Staff: Mentor

Multiply both sides by 2. arc tan x is a function of x.

Chet

6. May 4, 2014

### delsoo

i redo the question and dont know how to proceed here... any idea on how should i do next ? i dont know how to integreate arc tan x

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7. May 4, 2014

### Staff: Mentor

Integrate by parts. Do you remember how to take the derivative of arc tan x with respect to x?

Chet

8. May 4, 2014

### delsoo

sorry , i checked thru the syllabus, there's no deriative of arc tan x in it or maybe i can do it in other way? any other way?

9. May 4, 2014

### Staff: Mentor

Yes. Let y = arc tan x

Then tan y = x

Differentiating both sides with respect to x;

$$sec^2y\frac{dy}{dx}=1$$

Also, we have the trig identity: $tan^2y+1=sec^2y$

So, $sec^2y=1+x^2$

So, $$\frac{dy}{dx}=\frac{1}{1+x^2}$$

So, $$\frac{d(tan^{-1}x)}{dx}=\frac{1}{1+x^2}$$

Chet

10. May 4, 2014

### delsoo

thanks got the solution finally!