Question #284292

Verify that the given differential equation is an exact: 

(3x2+2y)dx+(x-4y)dy=0


1
Expert's answer
2022-01-04T11:26:00-0500

To verify if the given differential equation is exact,

Give the equation

(3x2+2y)dx+(x4y)dy=0(3x^{2}+2y)dx+(x-4y)dy=0

wherewhere

M=3x2+2yM=3x^{2}+2y and

N=x4yN=x-4y

For exactness, we know that

My=Nx\frac{∂M}{∂y}=\frac{∂N}{∂x}


Now, we shall find My\frac{∂M}{∂y}

My=2\frac{∂M}{∂y}=2

Also, we shall find Nx\frac{∂N}{∂x}

Nx=1\frac{∂N}{∂x}=1


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


Hence, the differential equation (3x2+2y)dx+(x4y)dy=0(3x^{2}+2y)dx+(x-4y)dy=0

Is not exact.

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