Misc 8

Transcript

Misc 8 Find the equation of the curve passing through the point 0 , 𝜋﷮4﷯﷯ whose differential equation is sin 𝑥 cos﷮𝑦 𝑑𝑥+ cos﷮𝑥 sin﷮𝑦 𝑑𝑦=0﷯﷯﷯ sin x cos y dx + cos x sin y dy = 0 sin x cos y dx = − cos x sin y dy sin ﷮𝑥﷯﷮ cos﷮𝑥﷯﷯ 𝑑𝑥 = − sin﷮𝑦﷯﷮ cos﷮𝑦﷯﷯ 𝑑𝑦 Integrating both sides ﷮﷮ sin﷮𝑥 𝑑𝑥﷯﷮ cos﷮𝑥﷯﷯﷯= ﷮﷮ −sin﷮𝑦 𝑑𝑦﷯﷮ cos﷮𝑦﷯﷯﷯ ∴ ﷮﷮ sin﷮𝑥﷯﷮𝑢﷯ × −𝑑𝑢﷮ sin﷮𝑥﷯﷯﷯= ﷮﷮ − sin﷮𝑦﷯﷮𝑣﷯ × −𝑑𝑣﷮ sin﷮𝑦﷯﷯﷯ ﷮﷮ 𝑑𝑣﷮𝑢﷯﷯=− ﷮﷮ 𝑑𝑢﷮𝑣﷯﷯ log 𝑢﷯=−𝑙𝑜𝑔 𝑣﷯+𝑐 log 𝑢﷯ + log 𝑣﷯ = c log 𝑢.𝑣﷯=𝑐 Put values of u and v. log cos﷮𝑥. cos﷮𝑦﷯﷯﷯= −c Since the curve passes through 0, 𝜋﷮4﷯﷯ Putting x = 0 and y = 𝜋﷮4﷯ in (1) log cos﷮ 0﷯﷯.cos⁡ 𝜋﷮4﷯﷯﷯=𝑐 log 1. 1﷮ ﷮2﷯﷯﷯=𝑐 ⇒ C = log 1﷮ ﷮2﷯﷯ Substitute value of C in (2) log cos﷮𝑥 cos﷮𝑦﷯﷯﷯=𝑐 log cos﷮𝑥. cos﷮𝑦﷯﷯﷯= log﷮ 1﷮ ﷮2﷯﷯﷯﷯ ∴ cos x. cos y = 1﷮ ﷮2﷯﷯ cos y = 1﷮ ﷮2﷯ cos﷮𝑥﷯﷯ cos y = 𝒔𝒆𝒄 𝒙﷮ ﷮𝟐﷯﷯