First slide

Types of solutions of differential equations

Question

$The solution of differential equation \frac{x d x - y d y}{x d y - y d x} = \sqrt{\frac{1 + x^{2} - y^{2}}{x^{2} - y^{2}}} is$

Moderate

A

$\sqrt{x^{2} - y^{2}} + \sqrt{x - y} + C$
B

$\sqrt{x^{2} - y^{2}} + \sqrt{x^{2} + y^{2}} + C$
C

$\sqrt{x^{2} - y^{2}} + \sqrt{1 + x^{2} - y^{2}} = |\frac{c (x + y)}{\sqrt{x^{2} - y^{2}}}|$
D

$\sqrt{x^{2} - y^{2}} + \sqrt{1 + x^{2} - y^{2}} = |\frac{c (x - y)}{\sqrt{x^{2} + y^{2}}}|$

Solution

$\begin{array}{l} Put x = r \sec θ y = r \tan θ \\ \Rightarrow \frac{d r}{\sec θ d θ} = \sqrt{1 + r^{2}} \\ \int \frac{d r}{\sqrt{1 + r^{2}}} = \int \sec θ d θ \\ \ln |r + \sqrt{1 + r^{2}}| = \ln | \sec θ + \tan θ | + \ln | c | \\ \Rightarrow \sqrt{x^{2} - y^{2}} + \sqrt{1 + x^{2} - y^{2}} =∣ \frac{c (x + y)}{\sqrt{x^{2} - y^{2}}} \end{array}$

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