Using a change in variables, Show that the boundary value problem:−ddx(p(x)y′)+q(x)y=f(x),a≤x≤b,y(a)=α,y(b)=βCan be transformed to the form:−ddw(p(w)z′)+q(w)z=F(w),0≤w≤1,z(0)=0,z(1)=0

Find out the answer to "Using a change in variables, Show that the..."

Destinee Bryan

2022-04-08

Using a change in variables, Show that the boundary value problem:

- \frac{d}{d x} (p (x) y^{'}) + q (x) y = f (x),

a \leq x \leq b,

y (a) = α,

y (b) = β

Can be transformed to the form:

- \frac{d}{d w} (p (w) z^{'}) + q (w) z = F (w),

0 \leq w \leq 1,

z (0) = 0,

z (1) = 0

mortaraskxnl

Beginner2022-04-09Added 12 answers

Step 1
Notice that you can define

ω (x)

\frac{x - a}{b - a} = \frac{ω (x) - 0}{1 - 0}

= ω (x) \Rightarrow x (ω)

= a + (b - a) ω,

and then

z (ω) = ω - \frac{y (x (ω)) - α}{β - α} .

Finally, perform the respective derivatives using this new variables. The differentiation chain rule will be your best friend in that deal!

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?