Suppose that y (a function of x) satisfies the following:
where dy/dx is the derivative of y. It is known also that y=1 when x=0.
Suppose that the function z is obtained from y as follows:
Use this to find an expression for y in terms of x.
Too difficult for me
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- Thread Starter
- 19-03-2004 10:00
- 19-03-2004 12:51
Applying the product rule to z = y * e^x gives
= dy/dx * e^x + y * e^x
= e^x * (dy/dx + y).
Since we know that dy/dx + y = x * e^(-x) it follows that
dz/dx = e^x * x * e^(-x) = x.
So z = (1/2)x^2 + constant.
When x = 0 we have y = 1 and hence z = 1. So z = (1/2)x^2 + 1.
So y = e^(-x) * z = e^(-x) * [(1/2)x^2 + 1].