Question

In Exercises 1–22, solve the differential equation.dy + x(2y - e^(x-x²))dx = 0

Accepted Answer

Rewrite the given differential equation in the form $$M(x,y)\,dx + N(x,y)\,dy = 0. $$Here, the equation is $$dy + x(2y - e$$^{x - x^{2}$$})\,dx = 0$$, which can be rearranged as $$x(2y - e$$^{x - x^{2}$$})\,dx + dy = 0. $$Identify the functions $$M(x,y)$$ and $$N(x,y)$$ from the equation: $$M(x,y) = x(2y - e$$^{x - x^{2}$$})$$ and $$N(x,y) = 1. $$Check if the differential equation is exact by verifying if $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}. $$Compute $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}. If $$the equation is not exact, look for an integrating factor that depends on either $$x or$$ $$y to $$make it exact. Determine the integrating factor by using the formula involving $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}. $$Once the equation is exact (either originally or after multiplying by the 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 a function of $$y$$, then differentiate with respect to $$y$$ and equate to $$N to $$find that function. Finally, write the implicit solution $$\Psi(x,y) = C.$$

In Exercises 1–22, solve the differential equation.

dy + x(2y - e^(x-x²))dx = 0

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Key Concepts

Differential Equations

Exact Differential Equations

Integrating Factor