In case an equation is in the form y = f(ax + by + c), i.e., the RHS is a linear function of x and y. We will use the su

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In Case An Equation Is In The Form Y F Ax By C I E The Rhs Is A Linear Function Of X And Y We Will Use The Su 1 (42.45 KiB) Viewed 61 times

In case an equation is in the form y = f(ax + by + c), i.e., the RHS is a linear function of x and y. We will use the substitution v = ax + by + c to find an implicit general solution. The right hand side of the following first order problem is a linear function of x and y. Use the substitution v = 3x - y + 1 to solve the initial value problem. y = 3e(3x-y+1) +3, y(0) = -1 We obtain the following separable equation in the variables x and v: v' = Solving this equation and transforming back to the variables x and y an implicit solution can be written in the form 3x + From this formula and the initial condition we can compute C = Finally we obtain the explicit solution of the initial value problem as y =