Find the integrating factor of the differential equation $\left( - 14 y ^ { 3 } + 2 \right) d x + \left( 21 x y ^ { 2 } + 7 x ^ { 3 } \right) d y = 0$ that is a function of $x$ or $y$ alone.
A) $- \frac { 21 } { x }$
B) $- \frac { 3 } { x }$
C) $\frac { 1 } { x ^ { 21 } }$
D) $\frac { 1 } { x ^ { 3 } }$
E) $\frac { 1 } { x ^ { 14 } }$

Question

Quizplus · Accepted Answer

The given differential equation can be written as $M(x,y)dx + N(x,y)dy = 0$, where $M = -14y^3 + 2$ and $N = 21xy^2 + 7x^3$. To find an integrating factor that is a function of $x$ or $y$ alone, we check if the equation is exact or can be made exact. The condition for exactness is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. Here, $\frac{\partial M}{\partial y} = -42y^2$ and $\frac{\partial N}{\partial x} = 21y^2 + 21x^2$, which are not equal, indicating the equation is not exact. To find an integrating factor that depends only on $x$ or $y$, we look for a function $\mu(x)$ or $\mu(y)$ such that $\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})$ is a function of $x$ alone or $y$ alone. Here, $\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}) = \frac{-42y^2 - (21y^2 + 21x^2)}{21xy^2 + 7x^3} = \frac{-21x^2}{21xy^2 + 7x^3}$, which simplifies to a function of $x$ alone. Thus, we seek an integrating factor $\mu(x)$ that makes the equation exact. The correct integrating factor that depends only on $x$ and makes the differential equation exact is $\frac{1}{x^3}$, which corresponds to choice D.

Find the Integrating Factor of the Differential Equation (−14y3+2)dx+(21xy2+7x3)dy=0\left( - 14 y ^ { 3 } + 2 \right) d x + \left( 21 x y ^ { 2 } + 7 x ^ { 3 } \right) d y = 0(−14y3+2)dx+(21xy2+7x3)dy=0

Find the Integrating Factor of the Differential Equation $\left( - 14 y ^ { 3 } + 2 \right) d x + \left( 21 x y ^ { 2 } + 7 x ^ { 3 } \right) d y = 0$