Questions: Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.) x dy/dx = 3x e^x - y + 6x^2

Solution Steps

The given differential equation is: \[ x \frac{d y}{d x} = 3 x e^{x} - y + 6 x^{2} \] To determine if it is exact, we first rewrite it in the standard form: \[ M(x, y) \, dx + N(x, y) \, dy = 0 \] Rearranging the given equation: \[ x \, dy = (3 x e^{x} - y + 6 x^{2}) \, dx \] \[ (3 x e^{x} - y + 6 x^{2}) \, dx - x \, dy = 0 \] Thus, we identify: \[ M(x, y) = 3 x e^{x} - y + 6 x^{2}, \quad N(x, y) = -x \]

A differential equation is exact if: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \] Compute \(\frac{\partial M}{\partial y}\): \[ \frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (3 x e^{x} - y + 6 x^{2}) = -1 \] Compute \(\frac{\partial N}{\partial x}\): \[ \frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (-x) = -1 \] Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the equation is exact.

To solve the exact differential equation, we find a function \(F(x, y)\) such that: \[ \frac{\partial F}{\partial x} = M(x, y), \quad \frac{\partial F}{\partial y} = N(x, y) \] Integrate \(M(x, y)\) with respect to \(x\): \[ F(x, y) = \int (3 x e^{x} - y + 6 x^{2}) \, dx = 3 \int x e^{x} \, dx - y \int dx + 6 \int x^{2} \, dx \] Using integration by parts for \(\int x e^{x} \, dx\): \[ \int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx = x e^{x} - e^{x} + C \] Thus: \[ F(x, y) = 3(x e^{x} - e^{x}) - y x + 2 x^{3} + g(y) \] Simplify: \[ F(x, y) = 3 x e^{x} - 3 e^{x} - y x + 2 x^{3} + g(y) \] Now, differentiate \(F(x, y)\) with respect to \(y\) and set it equal to \(N(x, y)\): \[ \frac{\partial F}{\partial y} = -x + g'(y) = -x \] This implies: \[ g'(y) = 0 \quad \Rightarrow \quad g(y) = C \] Thus, the general solution is: \[ F(x, y) = 3 x e^{x} - 3 e^{x} - y x + 2 x^{3} + C = 0 \] Rearranging: \[ 3 x e^{x} - 3 e^{x} - y x + 2 x^{3} = C \] Solve for \(y\): \[ y = \frac{3 x e^{x} - 3 e^{x} + 2 x^{3} - C}{x} \]

Final Answer