Answer to Question #284292 in Differential Equations for Aysu

Question

Verify that the given differential equation is an exact: 

(3x²+2y)dx+(x-4y)dy=0

Answer 1

To verify if the given differential equation is exact,

Give the equation

$(3x^{2}+2y)dx+(x-4y)dy=0$

$where$

$M=3x^{2}+2y$ and

$N=x-4y$

For exactness, we know that

$\frac{∂M}{∂y}=\frac{∂N}{∂x}$

Now, we shall find $\frac{∂M}{∂y}$

$\frac{∂M}{∂y}=2$

Also, we shall find $\frac{∂N}{∂x}$

$\frac{∂N}{∂x}=1$

Since, $\frac{∂M}{∂y} ≠ \frac{∂N}{∂x}$

Hence, the differential equation $(3x^{2}+2y)dx+(x-4y)dy=0$

Is not exact.