(1 point) 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 u

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1 Point 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 U 1 (76.56 KiB) Viewed 10 times

(1 point) 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 = (4x - 2y + 1) 5/6 +2, y(0) = 0 f(ax +by+c). To solve this problem we use the substitution v = ax + by + c which transforms the = is a function of a linear combination of x and y, i.e., y' equation into a separable equation. We obtain the following separable equation in the variables and v: v Solving this equation an implicit general solution in terms of a, u can be written in the form x+ Transforming back to the variables x and y the above equation becomes x+ Next using the initial condition y(0) = 0 we find C Then, after a little algebra, we can write the unique explicit solution of the initial value problem as y = = C. = C.