How can substitution be used to solve first-order equations?

[strawburry]

Homework Statement

Consider the equation y' = f(at + by + c)
where a, b, and c are constants. Show that the substitution x = at + by + c
changes the equation to the separable equation x' = a + bf(x).
Use this method to find the general solution of the equation y' = (y+t)^2

Homework Equations

n/a

The Attempt at a Solution

not sure where to begin :/

 

[rock.freak667]

The Attempt at a Solution

not sure where to begin :/

so x = at + by + c

x'=?

 

[strawburry]

mm so plug in the first equation into where by is?

 

[Dick]

Differentiate x with respect to t and solve for y' in terms of x'. Then put that expression for y' into your original equation. Now everything is in terms of x.

 

[strawburry]

thanks! i got how to do the first part, showing the substitution to show separable equation..

How do i continue on to second part of the question?? SOrryy I am kinda slow :(

 

[Dick]

Slow is ok, but your aren't helping yet. As rock.freak667 already asked, what is x'?

 

[strawburry]

x' = a + by' ??

 

[Dick]

x' = a + by' ??

Sure. Now solve that for y' and put it back into your original equation to eliminate y.

 

[strawburry]

y' = ( x' - a ) / b

then f(x) = (x'-a)/b

then x' = a + bf(x)

 

[Dick]

Well, that's that then, right?

 

[strawburry]

still not understanding
Use this method to find the general solution of the equation y' = (y+t)^2

 

[Dick]

still not understanding
Use this method to find the general solution of the equation y' = (y+t)^2

Try it. Substitute x=y+t. Do the same thing you just did to get an equation for x. What is it?

 

[strawburry]

x = y + t
x' = y' +1

y= x^2...x'-1= x^2

 

[Dick]

x = y + t
x' = y' +1

y= x^2...


x'-1= x^2

Nice. Ok, so dx/dt=(1+x^2). That's a separable ode.

 

[strawburry]

integral ( -x^2 dx) = integral (1)

- 1/3 (x^3) = t

x = cubed root (-3t) ?

 

[Dick]

integral ( -x^2 dx) = integral (1)

- 1/3 (x^3) = t

x = cubed root (-3t) ?

Oh, come on, that's just silly. dx/(1+x^2)=dt. Integrate both sides. I was sort of hoping you knew separable ODE's. Heard of them?

 

[strawburry]

haha just started learning them :P

 

[Dick]

haha just started learning them :P

Now's the time to use them.

 

[strawburry]

tytyty