Ex 9.3, 9 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.3, 9 For each of the differential equations in Exercises 1 to 10, find the general solution : 𝑑𝑦/𝑑𝑥=sin^(−1)⁡𝑥𝑑𝑦/𝑑𝑥=sin^(−1)⁡𝑥 𝑑𝑦 = sin^(−1)⁡𝑥 dx Integrating both sides ∫1▒〖𝒅𝒚 〗= ∫1▒〖〖𝐬𝐢𝐧〗^(−𝟏)⁡〖𝒙.𝟏 𝒅𝒙〗 〗 Integrating by parts, using formula ∫1▒〖𝑓 (𝑥)𝑔(𝑥)𝑑𝑥 〗= 𝑓(𝑥) ∫1▒〖𝑔(𝑥)𝑑𝑥 −∫1▒〖[𝑓′(𝑥)∫1▒𝑔(𝑥)𝑑𝑥] 𝑑𝑥 〗 〗 Take f(x) = sin−1 x and g(x) = 1 y = x 〖𝒔𝒊𝒏〗^(−𝟏) 𝒙 − ∫1▒𝒙/√(𝟏 − 𝒙^𝟐 ) dx Let t = 1 − x2 dt = −2xdx x dx = (−𝑑𝑡)/2 Hence, our equation becomes y = x sin−1 x − ∫1▒(−𝑑𝑡)/(2√𝑡) y = x sin−1 x + ∫1▒𝑑𝑡/(2√𝑡) y = x sin−1 x + 𝟏/𝟐 ∫1▒〖𝒕^((−𝟏)/𝟐) 𝒅𝒕〗 y = x sin−1 x + 𝟏/𝟐 𝒕^((−𝟏)/𝟐 + 𝟏)/((−𝟏)/𝟐 + 𝟏) + C y = x sin−1 x + 1/2 (𝑡^(1/2) )/((1/2) )+𝐶 y = x sin−1 x + √𝑡 + C Putting back value of t y = x sin−1 x + √(𝟏−𝒙^𝟐 ) + C y = sin−1 x ∫1▒〖𝟏 𝒅𝒙 −∫1▒[𝟏/√(𝟏 − 𝒙^𝟐 ) ∫1▒〖𝟏.𝒅𝒙 〗] 〗 dx