In Exercises 129–132 solve the initial value problem.
131. x dy - (y + √y)dx = 0, y(1) = 1

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Rewrite the given differential equation in the form \(M(x,y)\,dx + N(x,y)\,dy = 0\). Here, the equation is \(x\,dy - (y + \sqrt{y})\,dx = 0\), so we can write it as \(-(y + \sqrt{y})\,dx + x\,dy = 0\).

Check if the differential equation is exact by computing the partial derivatives \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\), where \(M = -(y + \sqrt{y})\) and \(N = x\).

If the equation is not exact, look for an integrating factor that depends on either \(x\) or \(y\) to make it exact. Consider the form of \(M\) and \(N\) to decide which variable the integrating factor might depend on.

Once the equation is exact, find the potential function \(\Psi(x,y)\) such that \(\frac{\partial \Psi}{\partial x} = M\) and \(\frac{\partial \Psi}{\partial y} = N\). Integrate \(M\) with respect to \(x\) and then determine the function of \(y\) by comparing with \(N\).

Use the initial condition \(y(1) = 1\) to solve for the constant of integration after finding the implicit solution \(\Psi(x,y) = C\).

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