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Can someone help me with ex 6e question 7a
(1+2x)dy/dx = x + 2y^2
Show the (1+2x) d3y/dx3 +4(1-y)d2y/dx2 = 4 (dy/dx)^2
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differentiate with respect to x twice (using product rule and implicit differentiation). When you differentiate y you get dy/dx. When you differentiate dy/dx you get d2y/dx2 (and so on).
Differentiating first time
u = 1+2x
u' = 2
v = dy/dx
v' = d2y/dx2
LHS: u'v + uv' = 2dy/dx + (1+2x)(d2y/dx2)
RHS: 1 + 4y(dy/dx)
so after differentiating with respect to x once, you get:
(1+2x)(d2y/dx2) + 2dy/dx = 1 + 4y(dy/dx)
Differentiating for the second time
If you differentiate with respect to x again, you get:
(1+2x)(d3y/dx3) + 4(d2y/dx2) = 4(dy/dx)^2 + 4y(d2y/dx2)
I did this in a similar way to the first time.
You can then rearrange this to get your show that answer.
This all comes under something called implicit differentiation, so if you need more help I'd search that on youtube or look it up in a text book :-) hope this helps