Example 27 - Chapter 9 Class 12 Differential Equations (Term 2)

Transcript

Example 27 Solve the differential equation (𝑥 𝑑𝑦−𝑦 𝑑𝑥)𝑦 𝑠𝑖𝑛(𝑦/𝑥)=(𝑦 𝑑𝑥+𝑥 𝑑𝑦)𝑥 cos⁡(𝑦/𝑥) (𝑥 𝑑𝑦−𝑦 𝑑𝑥)𝑦 𝑠𝑖𝑛(𝑦/𝑥)=(𝑦 𝑑𝑥+𝑥 𝑑𝑦)𝑥 cos⁡(𝑦/𝑥) 𝑥 𝑦 sin⁡〖(𝑦/𝑥)𝑑𝑦−𝑦^2 𝑠𝑖𝑛(𝑦/𝑥) 〗 𝑑𝑥=𝑥𝑦 cos⁡〖(𝑦/𝑥)𝑑𝑥+𝑥^2 〗 cos⁡(𝑦/𝑥)𝑑𝑦 [𝑥 𝑦 sin⁡〖(𝑦/𝑥)−𝑥^2 𝑐𝑜𝑠(𝑦/𝑥) 〗 ]𝑑𝑦=[𝑥𝑦 cos⁡〖(𝑦/𝑥)+𝑦^2 〗 sin⁡(𝑦/𝑥) ]𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑥𝑦 cos⁡〖(𝑦/𝑥) + 𝑦^2 sin⁡(𝑦/𝑥) 〗)/(𝑥𝑦 sin⁡〖(𝑦/𝑥) − 𝑥^2 𝑐𝑜𝑠(𝑦/𝑥)〗 ) Dividing numerator & denominator by x2 𝑑𝑦/𝑑𝑥 = (𝑦/𝑥 cos⁡〖(𝑦/𝑥) + (𝑦/𝑥)^2 cos⁡(𝑦/𝑥) 〗)/(𝑦/𝑥 sin⁡〖(𝑦/𝑥) − cos⁡(𝑦/𝑥) 〗 ) Putting y = vx. Differentiating w.r.t. x 𝑑𝑦/𝑑𝑥 = 𝑥 𝑑𝑣/𝑑𝑥 + v Putting value of 𝑑𝑦/𝑑𝑥 and y in (1) 𝑑𝑦/𝑑𝑥 = (𝑦/𝑥 cos⁡〖(𝑦/𝑥) + (𝑦/𝑥)^2 cos⁡(𝑦/𝑥) 〗)/(𝑦/𝑥 sin⁡〖(𝑦/𝑥) − cos⁡(𝑦/𝑥) 〗 ) v + (𝑥 𝑑𝑣)/𝑑𝑥=(𝑣𝑥/𝑥 cos⁡〖(𝑣𝑥/𝑥) + (𝑣^2 𝑥^2)/𝑥^2 sin⁡(𝑣𝑥/𝑥) 〗)/(𝑣𝑥/𝑥 sin⁡〖(𝑣𝑥/𝑥) − 𝑐𝑜𝑠(𝑣𝑥/𝑥)〗 ) v + (𝑥 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡𝑣 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡𝑣 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) − v x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡〖𝑣 − 𝑣(𝑣 sin⁡〖𝑣 − cos⁡〖𝑣)〗 〗 〗 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡〖𝑣 − 𝑣^2 sin 𝑣〗 + 𝑣 cos⁡𝑣 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) ( 𝑥 𝑑𝑣)/𝑑𝑥 = (2𝑣 cos⁡𝑣)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) ((𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗)/〖v cos〗⁡𝑣 )𝑑𝑣=2 𝑑𝑥/𝑥 ((𝑣 sin⁡𝑣)/〖v cos〗⁡𝑣 −cos⁡𝑣/〖v cos〗⁡𝑣 ) dv = 2 𝑑𝑥/𝑥 (tan⁡𝑣−1/𝑣) dv = 2 𝑑𝑥/𝑥 Integrating both sides ∫1▒〖(tan⁡〖𝑣 −1/𝑣〗 )𝑑𝑣=2∫1▒𝑑𝑥/𝑥〗 ∫1▒tan⁡〖𝑣 𝑑𝑣 − 〗 ∫1▒𝑑𝑣/𝑣 = 2 ∫1▒𝑑𝑥/𝑥 log |sec⁡𝑣 |−log⁡〖|𝑣|〗 = 2 log |𝑥| + log C log |sec⁡𝑣 |−log⁡〖|𝑣|〗 = log |𝑥^2 | + log C log |sec⁡𝑣/𝑣| = log 𝑥^2 + log C log |sec⁡𝑣/𝑣| = log 𝑥^2 𝑐 sec⁡𝑣/𝑣 = 𝑥^2 𝑐 Putting back value of v = 𝑦/𝑥 〖sec 〗⁡(𝑦/𝑥)/((𝑦/𝑥) ) = 𝑥^2 𝑐 sec⁡(𝑦/𝑥) = (𝑦/𝑥) 𝑥^2 𝑐 sec (𝒚/𝒙) = C xy (As log a – log b = log 𝑎/𝑏 & log a + log b = log ab)