Ex 9.4, 15 - Find equation of curve passing through (0, 0) - Ex 9.4

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.4, 15 Find the equation of curve passing through the point 0 , 0﷯ and whose differential equation is 𝑦﷮′﷯= 𝑒﷮𝑥﷯ sin﷮𝑥﷯ 𝑦﷮′﷯ = 𝑒﷮𝑥﷯ sin x 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮𝑥﷯ sin x dx 𝑑𝑦 = 𝑒﷮𝑥﷯ sin x dx Integrating both sides ﷮﷮𝑑𝑦﷯ = ﷮﷮𝑒𝑥 sin﷮𝑥 𝑑𝑥﷯﷯ y = ﷮﷮𝑒𝑥 sin﷮𝑥 𝑑𝑥﷯﷯ y = 𝑒﷮𝑥﷯ ﷮﷮ 𝑠𝑖𝑛 𝑥﷮ 𝑑𝑥﷯−﷯ ﷮﷮ ( 𝑒﷮𝑥﷯) ﷮﷮ sin﷮𝑥﷯𝑑𝑥 ﷯﷯𝑑𝑥﷯ y = 𝑒﷮𝑥﷯(− cos﷮𝑥﷯) − ﷮﷮(− cos﷮𝑥﷯)﷯ 𝑒﷮𝑥﷯ 𝑑𝑥 y = – 𝑒﷮𝑥﷯ 𝑐𝑜𝑠﷮𝑥﷯+ ﷮﷮ 𝑒﷮𝑥﷯ cos﷮𝑥﷯ 𝑑𝑥﷯ y = – 𝑒﷮𝑥﷯ 𝑐𝑜𝑠﷮𝑥﷯+ 𝑒﷮𝑥﷯ ﷮﷮ 𝑐𝑜𝑠 𝑥﷮ 𝑑𝑥﷯−﷯ ﷮﷮ ( 𝑒﷮𝑥﷯) ﷮﷮ cos﷮𝑥﷯𝑑𝑥 ﷯﷯𝑑𝑥﷯ y = – 𝑒﷮𝑥﷯ 𝑐𝑜𝑠﷮𝑥﷯+ 𝑒﷮𝑥﷯ 𝑠𝑖𝑛﷮𝑥﷯ − ﷮﷮ 𝑒﷮𝑥﷯ sin﷮𝑥﷯𝑑𝑥﷯ y = – 𝑒﷮𝑥﷯ 𝑐𝑜𝑠﷮𝑥﷯+ 𝑒﷮𝑥﷯ 𝑠𝑖𝑛﷮𝑥﷯ −𝑦 y + y = – 𝑒﷮𝑥﷯ 𝑐𝑜𝑠﷮𝑥﷯+ 𝑒﷮𝑥﷯ 𝑠𝑖𝑛﷮𝑥﷯ 2y = 𝑒﷮𝑥﷯( sin﷮𝑥﷯− cos﷮𝑥﷯) y = 1﷮2﷯ 𝑒﷮𝑥﷯ sin﷮𝑥﷯− cos﷮𝑥﷯﷯+ C Given curve passes through (0, 0) Putting x = 0, y = 0 in equation 0 = 1﷮2﷯ 𝑒﷮0﷯ ( sin﷮0﷯− cos﷮0﷯) + C 0 = 1﷮2﷯ (0 −1) + C 0 = −1﷮2﷯ + C C = 1﷮2﷯ Putting value of C in (1) y = 1﷮2﷯ 𝑒﷮𝑥﷯ sin﷮𝑥﷯− cos﷮𝑥﷯﷯ + 1﷮2﷯ y – 1﷮2﷯= 1﷮2﷯ 𝑒﷮𝑥﷯ sin﷮𝑥﷯− cos﷮𝑥﷯﷯ 2𝑦 − 1﷮2﷯= 1﷮2﷯( 𝑒﷮𝑥﷯ sin﷮𝑥﷯+ 𝑒﷮𝑥﷯ cos﷮𝑥﷯) 2y – 1 = ( 𝒆﷮𝒙﷯ 𝒔𝒊𝒏﷮𝒙﷯+ 𝒆﷮𝒙﷯ 𝒄𝒐𝒔﷮𝒙﷯)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.