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DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS. Note : Differential equations are equations containing a derivative. They can be solved by integration to obtain a general solution with +C. To obtain a specific solution requires additional information. Ex1. dy / dx + 7 = 2x. dffntl eqtn.

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DIFFERENTIAL EQUATIONS

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  1. DIFFERENTIAL EQUATIONS Note: Differential equations are equations containing a derivative. They can be solved by integration to obtain a general solution with +C. To obtain a specific solution requires additional information. Ex1 dy/dx + 7 = 2x dffntl eqtn dy/dx = 2x - 7 Integrate with respect to x y = (2x – 7) dx y = x2 – 7x + C General Soln

  2. Ex2 Given that dy/dx = x2 + 4 x2 and y = f(x) passes through (-2,2) then find the exact solution to this differential equation. ********* dy/dx = x2 + 4 x2 = x2 + 4 . x2 x2 = 1 + 4x-2 So y = (1 + 4x-2) dx At (-2,2) this becomes 2 = -2 – (-2) + C y = x + 4x-1 + C -1 ie C = 2 y = x – 4 + C x Solution is y = x – 4 + 2 x

  3. Ex3 (from Physics) Newton’s 1st law of motion states that v = u + at v – final velocity, u - initial velocity, a – acceleration, t- time In Physics the symbol for distance is s Velocity is basically speed Given that speed = difference in distance difference in time We now have v = ds/dt ctd

  4. So v = u + at becomes ds/dt = u + at Integrate with respect to t giving s =  (u + at) dt or s = ut + 1/2at2 + C Assuming that s, u, t, a are all zero simultaneously gives C = 0 and we now get s = ut + 1/2at2 which is Newton’s 2nd law of motion.

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