Ex 9.6, 11 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.6, 11 For each of the differential equation find the general solution : 𝑦 𝑑𝑥+ 𝑥− 𝑦﷮2﷯﷯𝑑𝑦=0 Step 1 : Put in form 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q or 𝑑𝑥﷮𝑑𝑦﷯ + P1 x = Q1, y dx + (x − y2) dy = 0 y dx = − (x − y2)dy 𝑑𝑦﷮𝑑𝑥﷯ = −𝑦﷮𝑥− 𝑦﷮2﷯﷯ This is not of the form 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q ∴ we find 𝑑𝑥﷮𝑑𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑦﷮2﷯ − 𝑥﷮𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = y − 𝑥﷮𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ + 𝑥﷮𝑦﷯ = y Step 2 : Find P1 and Q1 Comparing (1) with 𝑑𝑥﷮𝑑𝑦﷯ + P1 x = Q1 Where P1 = 1﷮𝑦﷯ & Q1 = y Step 3 : Find Integrating factor, IF = 𝑒﷮ ﷮﷮𝑝1 𝑑𝑦﷯﷯ = 𝑒﷮ ﷮﷮ 𝑑𝑦﷮𝑦﷯﷯﷯ = 𝑒﷮ log﷮𝑦﷯﷯ = y Step 4 : Solution is x (IF) = ﷮﷮ 𝑄1×𝐼𝐹﷯𝑑𝑦+𝑐﷯ xy = ﷮﷮𝑦×𝑦 𝑑𝑦+𝑐﷯ xy = ﷮﷮ 𝑦﷮2﷯ 𝑑𝑦+𝑐﷯ xy = 𝑦﷮3﷯﷮3﷯+𝐶 x = 𝑦﷮3﷯﷮3𝑦﷯+ 𝐶﷮𝑦﷯ x = 𝒚﷮𝟐﷯﷮𝟑﷯+ 𝑪﷮𝒚﷯