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Solve (dy)/(dx) = yf^(')(x) = f(x) f^(')...

Solve `(dy)/(dx) = yf^(')(x) = f(x) f^(')(x)`, where `f(x)` is a given integrable function of `x`.

Text Solution

Verified by Experts

The correct Answer is:
`log_(e)(1+y-f(x))+f(x)+c=0`
`(dy)/(dx)+yf^(')(x)=f(x)f^(')(x)`
or `(dy)/(dx)=[f(x)-y]f^(')(x)`
Put `f(x)-y=t`
`therefore f^(')(x) = (dy)/(dx)=(dt)/(dx)`
Then the given equation transforms to
`f^(')(x)-(dt)/(dx)=(dt)/(dx)`
or `int(dt)/(1-t)=intf^(')(x)dx`
or `-log[1+y-f(x)]+f(x)+c=0`
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