Question

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

Accepted Answer

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 $$\Psi(x,y) = C.$$

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

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Key Concepts

Initial Value Problem (IVP)

Separable Differential Equations

Substitution Method for Nonlinear Terms