Example 19 - Chapter 9 Class 12 Differential Equations

Transcript

Example 19 Find the general solution of the differential equation 𝑑𝑦﷮𝑑𝑥﷯−𝑦= cos﷮𝑥﷯ Differential equation is of the form 𝑑𝑦﷮𝑑𝑥﷯+𝑃𝑦=𝑄 where P = −1 & Q = cos x IF = e﷮ ﷮﷮𝑝𝑑𝑥﷯﷯ IF = e﷮− ﷮﷮1𝑑𝑥﷯﷯ IF = 𝑒﷮−𝑥﷯ Solution is y(IF) = ﷮﷮ 𝑄×𝐼𝐹﷯ 𝑑𝑥+𝑐﷯ 𝑦 𝑒﷮−𝑥﷯ = ﷮﷮ 𝑒﷮−𝑥﷯﷯ cos﷮𝑥+𝑐﷯ Let I = ﷮﷮ 𝑒﷮−𝑥﷯﷯ cos﷮𝑥 𝑑𝑥﷯ I = cos x ﷮﷮ 𝑒﷮−𝑥﷯𝑑𝑥﷯− ﷮﷮ − sin﷮𝑥 ﷮﷮ 𝑒﷮−𝑥﷯𝑑𝑥﷯﷯﷯𝑑𝑥﷯ I = −𝑒﷮−𝑥﷯cos x − ﷮﷮− sin﷮𝑥 (− 𝑒﷮−𝑥﷯﷯)﷯𝑑𝑥 I = −e−x cos x − ﷮﷮ 𝑒﷮−𝑥﷯ sin﷮𝑥 𝑑𝑥﷯﷯ I = −e−x cos x − sin﷮𝑥 ﷮﷮ 𝑒﷮−𝑥﷯𝑑𝑥﷯− ﷮﷮ (cos﷮𝑥 ﷮﷮ 𝑒﷮−𝑥﷯𝑑𝑥﷯)﷯﷯ dx ﷯﷯ I = −e−x cos x − − 𝑒﷮−𝑥﷯ sin﷮𝑥 − ﷮﷮− 𝑒﷮−𝑥﷯ cos﷮𝑥﷯𝑑𝑥﷯ ﷯﷯ I = −e−x cos x − − 𝑒﷮−𝑥﷯ sin﷮𝑥+ ﷮﷮ 𝑒﷮−𝑥﷯ cos﷮𝑥﷯𝑑𝑥﷯ ﷯﷯ I = −e−x cos x + 𝑒﷮−𝑥﷯ sin﷮𝑥− ﷮﷮ 𝒆﷮−𝒙﷯ 𝒄𝒐𝒔﷮𝒙﷯𝒅𝒙﷯ ﷯ I = e−x (sin x − cos x) − I 2I = e−x (sin x − cos x) I = 𝑒﷮−𝑥﷯﷮2﷯ (sin x − cos x) From (1) y 𝑒﷮−𝑥﷯ = ﷮﷮ 𝑒﷮−𝑥﷯ cos﷮𝑥+𝑐﷯﷯ y 𝑒﷮−𝑥﷯ = 𝑒﷮−𝑥﷯﷮2﷯ (sin x − cos x) + c y = 𝟏﷮𝟐﷯ (sin x − cos x) + c 𝒆﷮−𝒙﷯