Q3Mathematics II

Question

Q.3. If $r = x i + y j + z k$ , then show that $\nabla (1/ r) = - r / r^{3}$ .

Answer 1

Given the first-order differential equation:

(1 + y^{2}) d x = (tan^{- 1} y - x) d y

We systematically divide by $(1 + y^{2}) d y$ to explicitly rearrange it entirely into the standard linear form for $x$ as a dependent variable of $y$ (i.e., $\frac{d x}{d y} + P (y) x = Q (y)$ ):

\frac{d x}{d y} = \frac{tan ^{- 1} y}{1 + y ^{2}} - \frac{x}{1 + y ^{2}}

\frac{d x}{d y} + (\frac{1}{1 + y ^{2}}) x = \frac{tan ^{- 1} y}{1 + y ^{2}}

Here, the coefficient function $P (y)$ is exactly $\frac{1}{1 + y ^{2}}$ . The fundamental Integrating Factor (I.F.) is rigidly defined as $e^{\int P (y) d y}$ :

I.F. = e^{\int \frac{1}{1 + y ^{2}} d y}

Since the standard integral of $\frac{1}{1 + y ^{2}}$ is exactly $tan^{- 1} y$ :

I.F. = e^{t a n^{- 1} y}

Q3Mathematics II

Question

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