Question

In Exercises 1–22, solve the differential equation.x dy + (3y - x⁻² cos x) dx = 0, x \u003e 0

Accepted Answer

Rewrite the given differential equation in the form $$M(x,y)\,dx + N(x,y)\,dy = 0. $$Here, the equation is $$x\,dy + (3y - x$$^{-2}$$ \cos x)\,dx = 0$$, so we can write it as $$(3y - x$$^{-2}$$ \cos x)\,dx + x\,dy = 0$$ where $$M = 3y - x$$^{-2}$$ \cos x$$ and $$N = x. $$Check if the differential equation is exact by computing the partial derivatives $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}. If $$they are equal, the equation is exact; otherwise, it is not exact. If the equation is not exact, try to find an integrating factor that depends on either $$x or$$ $$y to $$make it exact. For this, calculate $$\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}$$ and check if it can be expressed as a function of $$x$$ alone or $$y$$ alone. Once the equation is exact (either originally or after multiplying by an integrating factor), 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 include an unknown function of $$y$$, then differentiate this result with respect to $$y$$ and set it equal to $$N to $$solve for the unknown function. Write the implicit solution as $$\Psi(x,y) = C$$, where $$C is a $$constant. This represents the general solution to the differential equation.

In Exercises 1–22, solve the differential equation.
x dy + (3y - x⁻² cos x) dx = 0, x > 0

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Key Concepts

Differential Equations and Their Forms

Exact Differential Equations

Integrating Factors

In Exercises 1–22, solve the differential equation.x dy + (3y - x⁻² cos x) dx = 0, x > 0