# Math Help - Quick problem I need a hand with please!

1. ## Quick problem I need a hand with please!

I have this question as an assignment, puzzled with one bit...

let y = Q(x) be any solution to the differential equation.

dy/dx = xy

Answer each of the following without solving this DE.

(i) In terms of x only, what is the slope of Q as it crosses the line y = x ?

(ii) What is the slope of the line y = x ?

(iii) " In the region x > 1, the graph of Q can only cross the line y = x from below" Is this statement true or false??

My attempt...

(i)= slope = dy/dx = xy = x . x = x2

(ii)= 1

(iii)= TRUE = but the only way i found it to be true is by solving the DE and graphing it which is not allowed by the question. Am i right firstly in saying it is true and if so how would I know this without solving DE??

Thanks alot!

Nappy

2. The answers to the first two questions are correct, now just use those to answer the third question

You know that when $x=y$ we have $\frac{dy}{dx}=x^{2}$, so when we consider what is happening of the line we know that $y=\frac{1}{3}x^{3}+c$

when you draw the graph of these equations you will see that it is true

3. Originally Posted by Nappy
I have this question as an assignment, puzzled with one bit...

let y = Q(x) be any solution to the differential equation.

dy/dx = xy

Answer each of the following without solving this DE.

(i) In terms of x only, what is the slope of Q as it crosses the line y = x ?

(ii) What is the slope of the line y = x ?

(iii) " In the region x > 1, the graph of Q can only cross the line y = x from below" Is this statement true or false??

My attempt...

(i)= slope = dy/dx = xy = x . x = x2

(ii)= 1

(iii)= TRUE = but the only way i found it to be true is by solving the DE and graphing it which is not allowed by the question. Am i right firstly in saying it is true and if so how would I know this without solving DE??

Thanks alot!

Nappy
Simpler, I think. You have already said that, on this line, $dy/dx= x^2$ which is never negative. That is, the graph must cross the line from below because it cannot be decreasing there.

4. Thanks guys,

think ive gotten myself a little confused though. Although the slope can never be negative and is always increasing could it not be above the line y = x. I solved for the curve and got

y = 1/3 x3 + c

Since this has infinitely many solutions, wouldnt a high constant of integration push it above the line making the statement false or am I losing the plot here??

Thanks again....