Ex 9.5, 12 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.5, 12 For each of the differential equation find the general solution : (𝑥+3𝑦^2 ) 𝑑𝑦/𝑑𝑥=𝑦(𝑦>0) Step 1 : Put In form 𝑑𝑦/𝑑𝑥 + py = Q or 𝑑𝑥/𝑑𝑦 + P1x = Q1 (𝑥+3𝑦^2 ) 𝑑𝑦/𝑑𝑥=𝑦 𝑑𝑦/𝑑𝑥 = 𝑦/(𝑥+3𝑦^2 ) This is not of the form 𝑑𝑦/𝑑𝑥 + Py = Q ∴ We need to find 𝒅𝒙/𝒅𝒚 𝑑𝑥/𝑑𝑦 = (𝑥 + 3𝑦^2)/𝑦 𝒅𝒙/𝒅𝒚 = 𝒙/𝒚 + (𝟑𝒚^𝟐)/𝒚 Step 2 : Find P1 and Q1 Comparing with 𝑑𝑦/𝑑𝑥 + P1x = Q1 where P1 = (−𝟏)/𝒚 & Q1 = 3y Step 3 : Finding Integrating factor IF = 𝒆^(∫1▒𝒑_𝟏 𝒅𝒚) IF = 𝑒^(∫1▒(−1)/𝑦 𝑑𝑦" " ) IF = e−log y IF = 𝑒^log⁡〖𝑦^(−1) 〗 IF = y−1 IF = 𝟏/𝒚 Step 4 : Solution of the equation Solution is x(IF) = ∫1▒〖(𝑄1×𝐼𝐹)𝑑𝑦+𝐶〗 x(1/𝑦)=∫1▒〖3𝑦×1/𝑦 𝑑𝑦+𝐶〗 𝑥/𝑦 = 3∫1▒〖𝑑𝑦+𝐶〗 𝑥/𝑦 = 3𝑦+𝐶 𝒙 = 𝟑𝒚^𝟐+𝑪y