Solve the system for the exact special solution y = y(x): (keep the fraction and the square root without decimals.) 1. ydx + x[ in(x) - in(y) - 1]dy = 0 and y(1) = e for in(e) = 1.

Assume a solution of the form . Differentiating both sides gives

with and .

Divide both sides by and we have

Notice that

so the ODE is exact. Now we can look for a solution with

Integrating the first PDE with respect to gives

and differentiating this with respect to gives

So this ODE has general solution

Given that , we have

so the particular solution is

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step-by-step explanation: