Ex 9.6, 1 - Find general solution: dy/dx + 2y = sin x - Ex 9.6

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.6, 1 For each of the differential equation given in Exercises 1 to 12, find the general solution : 𝑑𝑦﷮𝑑𝑥﷯+2𝑦=𝑠𝑖𝑛𝑥 Step 3: Find integrating factor, I.F. I.F. = e﷮ ﷮﷮𝑝𝑑𝑥﷯﷯ I.F. = 𝑒﷮ ﷮﷮2𝑑𝑥﷯﷯= 𝑒﷮2𝑥﷯ So, I.F = 𝑒﷮2𝑥﷯ Step 4 : Solution of the equation y × I.F = ﷮﷮𝑄×𝐼.𝐹.𝑑𝑥+𝑐 ﷯ Putting values, 𝑦× 𝑒﷮2𝑥﷯ = ﷮﷮ sin﷮𝑥.𝑒 .﷮2𝑥﷯﷯.𝑑𝑥+𝑐﷯ 𝐿𝑒𝑡 𝐼= ﷮﷮ sin﷮𝑥. 𝑒﷮2𝑥﷯﷯.𝑑𝑥 ﷯ = sin x ﷮﷮ 𝑒﷮2𝑥﷯.𝑑𝑥− ﷮﷮ 𝑑﷮𝑑𝑥﷯ sin﷮𝑥﷯ ﷮﷮ 𝑒﷮2𝑥﷯𝑑𝑥 ﷯﷯ ﷯ ﷯ = sin x 𝑒﷮2﷯ − ﷮﷮ cos﷮𝑥﷯﷯ 𝑒﷮2𝑥﷯﷮2﷯ dx = 1﷮2﷯ sin﷮𝑥 𝑒﷮2𝑥﷯﷯− 1﷮2﷯ cos﷮𝑥﷯ ﷮﷮ 𝑒﷮2𝑥﷯﷯𝑑𝑥 − ﷮﷮ 𝑑﷮𝑑𝑥﷯﷯ cos﷮𝑥﷯ ﷮﷮ 𝑒﷮2𝑥﷯﷯𝑑𝑥 ﷯dx = 1﷮2﷯ sin﷮𝑥 𝑒﷮2𝑥﷯﷯− 1﷮2﷯ cos﷮𝑥﷯ ﷮﷮ 𝑒﷮2𝑥﷯﷮2﷯﷯ − ﷮﷮(−sin x)﷯ ﷮﷮ 𝑒﷮2𝑥﷯﷮2﷯﷯𝑑𝑥 ﷯ = 1﷮2﷯ sin﷮𝑥 𝑒﷮2𝑥﷯﷯− 1﷮2﷯ cos﷮𝑥﷯ 𝑒﷮2𝑥﷯﷮2﷯+ 1﷮2﷯ ﷮﷮𝐬𝐢𝐧 𝐱 𝒆﷮𝟐𝒙﷯ 𝒅𝒙﷯﷯ = 1﷮2﷯ sin﷮𝑥 𝑒﷮2𝑥﷯﷯− 1﷮2﷯ cos﷮𝑥﷯ 𝑒﷮2𝑥﷯﷮2﷯+ 1﷮2﷯ 𝑰﷯ + C I = 1﷮2﷯ sin x 𝑒﷮2𝑥﷯ − 1﷮4﷯ cos x 𝑒﷮2𝑥﷯ − 1﷮4﷯ I + C I + 1﷮4﷯ I = 1﷮4﷯ 2 sin﷮𝑥 𝑒﷮2𝑥﷯ − cos﷮𝑥 𝑒﷮2𝑥﷯﷯﷯﷯ + C 5𝐼﷮4﷯ = 𝑒﷮2𝑥﷯﷮4﷯ 2 sin﷮𝑥− cos﷮𝑥﷯﷯﷯ + C 𝐼 = 𝑒﷮2𝑥﷯﷮5﷯ 2 sin﷮𝑥− cos﷮𝑥﷯﷯﷯ + C Now, Putting value of I in (2) y 𝑒﷮2𝑥﷯ = 𝑒﷮2𝑥﷯﷮5﷯ 2 sin﷮𝑥 − cos﷮𝑥﷯﷯﷯ + C y = 𝟏﷮𝟓﷯ 𝟐 𝐬𝐢𝐧﷮𝒙 − 𝐜𝐨𝐬﷮𝒙﷯﷯﷯+𝑪 𝒆﷮−𝟐𝒙﷯

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