Question 34 (A) - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 34 (A) Solve the differential equation: 𝑦+𝑑/𝑑𝑥(𝑥𝑦)=𝑥(sin 𝑥+𝑥)Now, our equation is 𝑦+𝑑/𝑑𝑥(𝑥𝑦)=𝑥(sin 𝑥+𝑥) Using product formula 𝑦+(𝑑(𝑥))/𝑑𝑥 𝑦+𝑥 𝑑𝑦/𝑑𝑥=𝑥(sin 𝑥+𝑥) 𝑦+𝑦+𝑥 𝑑𝑦/𝑑𝑥=𝑥(sin 𝑥+𝑥) 𝑥 𝑑𝑦/𝑑𝑥+2𝑦=𝑥(sin 𝑥+𝑥) Diving both sides by x 𝑥/𝑥 \ × 𝑑𝑦/𝑑𝑥+2𝑦/𝑥=𝑥(sin 𝑥+𝑥)/𝑥 𝑑𝑦/𝑑𝑥+2𝑦/𝑥=(sin 𝑥+𝑥) Comparing with 𝑑𝑦/𝑑𝑥 + Py = Q P = 𝟐/𝒙 & Q = (𝒔𝒊𝒏 𝒙+𝒙) Finding Integrating factor (IF) IF = e^∫1▒𝑝𝑑𝑥 = 𝒆^∫1▒〖𝟐/𝒙 𝒅𝒙〗 = e^(2∫1▒〖1/𝑥 𝑑𝑥〗) = 𝒆^(𝟐 𝒍𝒐𝒈⁡|𝒙| ) = e^log⁡〖𝑥^2 〗 = 𝒙^𝟐 Solution of differential equation is y × IF = ∫1▒〖𝑄.𝐼𝐹 𝑑𝑥〗 Putting values y × x2 = ∫1▒〖(𝒔𝒊𝒏 𝒙+𝒙) 𝒙^𝟐 𝒅𝒙 〗 yx2 = ∫1▒〖𝑠𝑖𝑛⁡𝑥 × 𝑥^2 〗 𝑑𝑥+∫1▒𝒙^𝟑 𝒅𝒙 yx2 = ∫1▒〖𝑠𝑖𝑛⁡𝑥 × 𝑥^2 〗 𝑑𝑥+𝑥^4/4+𝐶 yx2 = ∫1▒〖𝒙^𝟐 𝒔𝒊𝒏⁡𝒙 〗 𝒅𝒙+𝒙^𝟒/𝟒+𝑪 Evaluating ∫1▒〖𝒙^𝟐 𝒔𝒊𝒏⁡𝒙 〗 𝒅𝒙 separately ∫1▒〖𝑥^2 sin⁡𝑥 〗 𝑑𝑥 We know that ∫1▒〖𝑓(𝑥) 𝑔⁡(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓^′ (𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = x2 and g(x) = sin x ∫1▒〖𝑥^2 sin⁡𝑥 〗 𝑑𝑥=𝒙^𝟐 ∫1▒𝐬𝐢𝐧⁡𝒙 𝒅𝒙−∫1▒(𝒅(𝒙^𝟐 )/𝒅𝒙 ∫1▒〖𝒔𝒊𝒏⁡𝒙 𝒅𝒙〗) 𝒅𝒙 = − 𝑥^2 cos⁡𝑥 − ∫1▒〖2𝑥 × −cos⁡〖𝑥 𝑑𝑥〗 〗 = − 𝑥^2 cos⁡𝑥+2 ∫1▒〖𝒙 𝒄𝒐𝒔⁡𝒙 〗 𝒅𝒙+𝐶 Applying by parts again in ∫1▒〖𝒙 𝒄𝒐𝒔⁡𝒙 〗 = − 𝑥^2 cos⁡𝑥+2 [𝒙∫1▒𝒄𝒐𝒔⁡𝒙 𝒅𝒙−∫1▒(𝒅𝒙/𝒅𝒙 ∫1▒〖𝒄𝒐𝒔⁡𝒙 𝒅𝒙〗) 𝒅𝒙]+𝐶 = − 𝑥^2 cos⁡𝑥+2 [𝒙 𝒔𝒊𝒏 𝒙−∫1▒〖𝟏 × 𝒔𝒊𝒏 𝒙〗 𝒅𝒙]+𝐶 = − 𝑥^2 cos⁡𝑥+2 [𝑥 𝑠𝑖𝑛 𝑥−∫1▒〖𝒔𝒊𝒏 𝒙〗 𝒅𝒙]+𝐶 = − 𝑥^2 cos⁡𝑥+2 [𝑥 𝑠𝑖𝑛 𝑥−(−𝑐𝑜𝑠 𝑥) ]+𝐶 = − 𝒙^𝟐 𝒄𝒐𝒔⁡𝒙+𝟐 [𝒙 𝒔𝒊𝒏 𝒙+𝐜𝐨𝐬⁡𝒙 ]+𝑪 Putting value of ∫1▒〖𝑥^2 sin⁡𝑥 〗 𝑑𝑥 in (1) yx2 = ∫1▒〖𝑥^2 𝑠𝑖𝑛⁡𝑥 〗 𝑑𝑥+𝑥^4/4+𝐶 yx2 = − 𝒙^𝟐 𝒄𝒐𝒔⁡𝒙+𝟐 [𝒙 𝒔𝒊𝒏 𝒙+𝐜𝐨𝐬⁡𝒙 ]+𝒙^𝟒/𝟒+𝑪 Which is the required solution