1. ## Picards Iterates

Needing a little help with this problem

Calculate the picards equation y' - y = x^2 with the condition y(0) = -1

First Step:
(Find Y prime)

y' = x^2 + y

y(0) = -1

then

Y1(x) = -1 + Integral from {0 to X} (t^2 - 1)dt
= -1 - x + t^3 / 3

Dont know if im doing this correct and what to do next.

2. Originally Posted by mathlg
Needing a little help with this problem

Calculate the picards equation y' - y = x^2 with the condition y(0) = -1

First Step:
(Find Y prime)

y' = x^2 + y

y(0) = -1

then

Y1(x) = -1 + Integral from {0 to X} (t^2 - 1)dt
= -1 - x + t^3 / 3

Dont know if im doing this correct and what to do next.

Where did the "t" come from?

Let's start over.

y' - y = x^2, y(0) = -1

Solve the homogeneous equation:
yh' - yh = 0

The characteristic equation is:
m - 1 = 0 ==> m = 1

Thus
yh(x) = Ae^{x}

Now we need a particular solution:
yp' - yp = x^2

Try
yp(x) = Bx^2 + Cx + D
yp'(x) = 2Bx + C

So
[2Bx + C] - [Bx^2 + Cx + D] = x^2

-Bx^2 + (2B - C)x + (C - D) = x^2

Thus
-B = 1
2B - C = 0
C - D = 0

Or B = -1, C = -2, and D = -2

So yp(x) = -x^2 - 2x - 2

Thus the solution to
y' - y = x^2
is
y(x) = yh(x) + yp(x) = Ae^{x} - x^2 - 2x - 2

We know that y(0) = -1, so
y(0) = Ae^{0} - (0)^2 - 2(0) - 2 = -1

A - 2 = -1

A = 1

Thus
y(x) = e^{x} - x^2 - 2x - 2
fits the bill.

-Dan

3. I think I understand how you got that. I think I messed up with the question. I forgot to write

Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

and it was given that y'(x) = x^2 +y and y(0)= -1

I was going off memory.

4. Not sure if im starting off the problem right.

Y(1) = -1 + integral from {0, X} (x^2 - 1)
= -1 + x^3 / 3 - 1x

Not sure how to get to Y2????

5. Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

and it was given that y'(x) = x^2 +y and y(0)= -1

Y1(t)= -1+ x^3/3- x

y'= x^2- 1+ x^3/3 - x.

For Y2 do I Integrate again Y2(0)= -1 ???

This correct??

Just want to see if my answer is correct now.

Y1(x) = y(0) + int(0,x) t^2 + Y0(t) dt
= -1 + int(0,x) t^2 - 1 dt
= -1 + x^3/3 - x
= x^3/3-x-1.

Y2(x) = y(0) + int(0,x) t^2 + Y1(t) dt
= -1 + int(0,x) (t^2 + t^3/3 - t - 1) dt
= x^4/12 + x^3/3 - x^2/2 - x - 1 .

Y3(x) = y(0) + int(0,x) t^2 + Y2(t) dt
= -1 + int(0,x) (t^2 + t^4/12 + t^3/3 - t^2/2 - t - 1)dt
= -1 + int(0,x) (t^4/12 + t^3/3 + t^2/2 - t - 1)dt
= x^5/60 + x^4/12 + x^3/6 - x^2/2 - x - 1

Y4(x) = y(0) + int(0,x) t^2 + Y3(t) dt
= -1 + int(0,x) (t^2 + t^5/60 + t^4/12 + t^3/6 - t^2/2 - t - 1) dt
= -1 + int(0,x) (t^5/60 + t^4/12 + t^3/6 - t^2/2 - t - 1)dt
= x^6/360 + x^5/60 + x^4/12 + x^3/6 - x^2/2 - x - 1