1 Point 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 1 (45.22 KiB) Viewed 12 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 A And Y We Will Use 2 (15.28 KiB) Viewed 12 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 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' = (3x - 2y + 1) 5/6 +/, y(0) = 0 is a function of a linear combination of x and y, i.e., y'= f(ar+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 and v v Solving this equation an implicit general solution in terms of a, v can be written in the form = C. x+ Transforming back to the variables and y the above equation becomes 2+ Next using the initial condition y(0) = 0 we find C = C.
Then, after a little algebra, we can write the unique explicit solution of the initial value problem as
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