Ex 9.3, 3 - Form differential equation: y = a e3x + b e-2x - Formation of Differntial equation when general solution given


  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.3, 3 Form a differential equation representing the given family of curves by eliminating arbitrary constants 𝑎 and 𝑏. 𝑦=𝑎 𝑒﷮3𝑥﷯+𝑏 𝑒﷮−2𝑥﷯ 𝑦=𝑎 𝑒﷮3𝑥﷯+𝑏 𝑒﷮−2𝑥﷯ ∴ Differentiating Both Sides w.r.t. 𝑥 𝑑𝑦﷮𝑑𝑥﷯= 𝑑﷮𝑑𝑥﷯ 𝑎 𝑒﷮3𝑥﷯+𝑏 𝑒﷮−2𝑥﷯﷯ =𝑎 𝑒﷮3𝑥﷯ 3﷯+𝑏 𝑒﷮−2𝑥﷯ −2﷯ =3𝑎 𝑒﷮3𝑥﷯−2𝑏 𝑒﷮−2𝑥﷯ ∴ 𝑦﷮′﷯=3𝑎 𝑒﷮3𝑥﷯−2𝑏 𝑒﷮−2𝑥﷯ Again differentiating w.r.t. 𝑥 𝑦﷮′′﷯= 𝑑﷮𝑑𝑥﷯ 3𝑎 𝑒﷮3𝑥﷯−2𝑏 𝑒﷮−2𝑥﷯﷯ 𝑦﷮′′﷯=3𝑎 𝑒﷮3𝑥﷯ 3﷯−2𝑏 𝑒﷮−2𝑥﷯ −2﷯ ∴ 𝑦﷮′′﷯=9𝑎 𝑒﷮3𝑥﷯+4𝑏 𝑒﷮−2𝑥﷯ Subtracting (2) From (1) 𝑦﷮′′﷯− 𝑦﷮′﷯=9𝑎 𝑒﷮3𝑥﷯+4𝑏 𝑒﷮−2𝑥﷯−3𝑎 𝑒﷮3𝑥﷯+2𝑏 𝑒﷮−2𝑥﷯ 𝑦﷮′′﷯− 𝑦﷮′﷯=6𝑎 𝑒﷮3𝑥﷯+6𝑏 𝑒﷮−2𝑥﷯ 𝑦﷮′′﷯− 𝑦﷮′﷯=6(𝑎 𝑒﷮3𝑥﷯+𝑏 𝑒﷮−2𝑥﷯) 𝑦﷮′′﷯− 𝑦﷮′﷯=6y 𝒚﷮′′﷯− 𝒚﷮′﷯−𝟔𝒚=𝟎 is the required differential equation.

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