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 (39.73 KiB) Viewed 52 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 Su 2 (18.61 KiB) Viewed 52 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 y = (7x - 3y + 1) 5/6 + , 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 x, u can be written in the form x+ = C. Transforming back to the variables x and y the above equation becomes x+ = C. 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 =
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