1 point) in case an equation is in the form y\' general solution. f(az + by + c)

Question

1 point) in case an equation is in the form y' general solution. f(az + by + c), ie., the RHs is a linear function of z and Y. We will use the substitution u az + by + c to find an implicit The right hand side of the following first order problem is a linear function of r and y. Use the substitution v- 2x -y + 2 to solve the initial value problem. y' = 2e(2x V+ 2) + 2, y(0)--2 We obtain the following separable equation in the variables z and t: t/ = Solving this equation and transforming back to the variables r and y an implicit solution can be written in the form 2x+ =C From this formula and the initial condition we can compute C Finally we obtain the explicit solution of the initial value problem as

Explanation / Answer

v = 2x- y+ 2
v' = 2 - y'
hence
y' =2 e^(2x-y+2) + 2
2 - v'= 2 e^v + 2
v' = -2e^v

v = -2 e^v + c
2x-y+2 = -2e^(2x-y+2 ) + c
2x +   (2-y + 2e^(2x-y+2 ) ) = C
y(0) = -2
2 +2 + 2 e^(4)) = C
C = 4 + 2 e^4

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