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 (42.48 KiB) Viewed 55 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 u = 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 u = x - y to solve the initial value problem. y = e(x-y) + 1, y(2) = 2 We obtain the following separable equation in the variables x and u: u' = Solving this equation and transforming back to the variables x and y we arrive at the implicit solution = -x + C Finally we obtain the explicit solution of the initial value problem as y =