Misc 10 - Chapter 9 Class 12 Differential Equations

Transcript

Misc 10 Solve the differential equation [𝑒^(−2√𝑥)/√𝑥−𝑦/√𝑥] 𝑑𝑥/𝑑𝑦=1(𝑥≠0)[𝑒^(−2√𝑥)/√𝑥−𝑦/√𝑥] 𝑑𝑥/𝑑𝑦=1 𝑒^(−2√𝑥)/√𝑥− 𝑦/√𝑥 =𝑑𝑦/𝑑𝑥 𝒅𝒚/𝒅𝒙 + 𝒚/√𝒙 = 𝒆^(−𝟐√𝒙)/√𝒙 Differential equation is of the form 𝑑𝑦/𝑑𝑥 + Py = Q where P = 𝟏/√𝒙 & Q = 𝒆^(−𝟐√𝒙)/√𝒙 IF. = e∫1▒"pdx" Finding ∫1▒〖𝑷 𝒅𝒙〗 ∫1▒〖𝑷 𝒅𝒙=∫1▒𝒅𝒙/√𝒙 〗 ∫1▒〖𝑃 𝑑𝑥=∫1▒〖𝑥^((−1)/2) 𝑑𝑥〗 〗 ∫1▒〖𝑃 𝑑𝑥=∫1▒〖(𝑥 (−1)/2 + 1)/((−1)/2 + 1) 𝑑𝑥〗 〗 ∫1▒〖𝑃 𝑑𝑥=2𝑥^(1/2) 〗= 2√𝑥 ∴ IF = 𝒆^(𝟐√𝒙) Solution is y (IF) = ∫1▒〖(𝑄×𝐼𝐹)𝑑𝑥+𝑐〗 y𝒆^(𝟐√𝒙) = ∫1▒〖(𝒆^(−𝟐√𝒙)/√𝒙×𝒆^(𝟐√𝒙) )𝒅𝒙+𝒄〗 y𝑒^(2√𝑥) = ∫1▒〖𝑑𝑥/√𝑥+𝑐〗 y𝑒^(2√𝑥) = ∫1▒〖1/√𝑥 𝑑𝑥+𝑐〗 y𝑒^(2√𝑥) = ∫1▒〖𝑥^((−1)/2) 𝑑𝑥+𝑐〗 y𝑒^(2√𝑥) = ∫1▒〖(𝑥^(−1/2 + 1) )/(−1/2 + 1)+𝑐〗 "y" 𝑒^(2√𝑥) " = "2𝑥^(1/2) + C "y" 𝒆^(𝟐√𝒙) " = "2√𝒙 + C