Answer Happy

In case an equation is in the form y' f(ax +by+c), i.e., the RHS is a linear function of a 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 y' = (7x − 5y + 4)³ + }, y(0) = 0 is a function of a linear combination of x and y, i.e., y' = f(ax +by+c). To solve this problem we use the substitution v = ax + by + c which transforms the equation into a separable equation. We obtain the following separable equation in the variables x and v: v = Solving this equation an implicit general solution in terms of a, v can be written in the form Next using the initial condition y(0) = 0 we find C= x+ Transforming back to the variables x and y we obtain an implicit solution x+ Then, after a little algebra, we can write the unique explicit solution of the initial value problem as y = = C. = C.