Ex 9.5, 2 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.5, 2 For each of the differential equation , find the𝑑𝑦/𝑑𝑥+3𝑦=𝑒^(−2𝑥) Step 1: Put in form 𝑑𝑦/𝑑𝑥 + Py = Q 𝒅𝒚/𝒅𝒙 + 3y = 𝒆^(−𝟐𝒙) Step 2: Find P and Q by comparing, we get 𝑷=𝟑 and Q = 𝒆^(−𝟐𝒙) Step 3 : Find Integrating factor, I.F. I.F. = 𝑒^∫1▒𝑝𝑑𝑥 I.F. = 𝑒^∫1▒3𝑑𝑥 general solution : 𝑑𝑦/𝑑𝑥+3𝑦=𝑒^(−2𝑥) I.F. = 𝒆^𝟑𝒙 Step 4 : Solution of the equation y × I.F. = ∫1▒〖𝑄×𝐼.𝐹. 𝑑𝑥+𝑐〗 Putting values y × e3x = ∫1▒𝒆^(−𝟐𝒙 + 𝟑𝒙) ,dx + 𝒄 ye3x = ∫1▒𝑒^(𝑥 ) dx + 𝑐 ye3x = 𝑒^(𝑥 ) dx + 𝑐 Dividing by 𝑒^(3𝑥 ) y = e–2x + Ce–3x