Ex 9.3, 5 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.3, 5 For each of the differential equations in Exercises 1 to 10, find the general solution : (𝑒^𝑥+𝑒^(−𝑥) )𝑑𝑦−(𝑒^𝑥−𝑒^(−𝑥) )𝑑𝑥=0 (𝑒^𝑥+𝑒^(−𝑥) )𝑑𝑦−(𝑒^𝑥−𝑒^(−𝑥) )𝑑𝑥=0 (𝑒^𝑥+𝑒^(−𝑥) )𝑑𝑦 = (𝑒^𝑥−𝑒^(−𝑥) )𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑒^𝑥 − 𝑒^(−𝑥))/(𝑒^𝑥 + 𝑒^(−𝑥) ) dx 𝒅𝒚 = (𝒆^𝒙 − 𝒆^(−𝒙))/(𝒆^𝒙 + 𝒆^(−𝒙) ) dx Integrating both sides. ∫1▒𝑑𝑦 = ∫1▒(𝑒^𝑥 − 𝑒^(−𝑥))/(𝑒^𝑥 + 𝑒^(−𝑥) ) dx 𝒚 = ∫1▒(𝒆^𝒙 − 𝒆^(−𝒙))/(𝒆^𝒙 + 𝒆^(−𝒙) ) dx Let t = 𝒆^𝒙+𝒆^(−𝒙) 𝑑𝑡/𝑑𝑥 = (𝑒^𝑥−𝑒^(−𝑥) ) dx = 𝒅𝒕/(𝒆^𝒙 − 𝒆^(−𝒙) ) Putting value of t and dt in (1) ∫1▒𝑑𝑦 = ∫1▒(𝑒^(𝑥 )−〖 𝑒〗^(−𝑥))/𝑡 𝑑𝑡/(𝑒^𝑥 − 𝑒^(−𝑥) ) . ∫1▒𝑑𝑦 = ∫1▒〖𝑑𝑡/𝑡 " " 〗 y = log |𝒕|+𝒄 Putting back t = 𝑒^𝑥−𝑒^(−𝑥) y = log |𝑒^𝑥−𝑒^(−𝑥) | + C As 𝒆^𝒙−𝒆^(−𝒙) > 0 So, its always positive Removing the modulus y = log (𝒆^𝒙−𝒆^(−𝒙)) + C