Example 21 - Chapter 9 Class 12 Differential Equations

Transcript

Example 21 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