First slide

Methods of solving first order first degree differential equations

Question

$The solution of \sec^{2} θ d θ + \tan θ (1 - r \tan θ) d r = 0 is$

Moderate

A

$\tan θ = r - 1 + c e^{r}$
B

$\tan θ = r + c e^{- r}$
C

$\cot θ = r - 1 + c e^{r}$
D

$\cot θ = r + c e^{- r}$

Solution

$\begin{array}{l} \frac{d θ}{d r} + \frac{\tan θ}{\sec^{2} θ} = \frac{r \tan^{2} θ}{\sec^{2} θ} \\ \cos e c^{2} θ \frac{d θ}{d r} + \cot θ = r \\ Put \cot θ = u \Rightarrow - {cosec}^{2} θ d θ = d u \\ \Rightarrow \frac{d u}{d r} - u = - r \\ I.F = e^{- r} G.S. is u e^{- r} = - r \int e^{- r} d r \Rightarrow \cot θ = r - 1 + c e^{r} \end{array}$

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