In Case An Equation Is In The Form Y F Ar By C I E The Rhs Is A Linear Function Of X And Y We Will Use The Substi 1 (38.74 KiB) Viewed 15 times
In Case An Equation Is In The Form Y F Ar By C I E The Rhs Is A Linear Function Of X And Y We Will Use The Substi 2 (8.97 KiB) Viewed 15 times
In case an equation is in the form y'= f(ar+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' (6x 5y + 3) +; = y(0) = 0 is a function of a linear combination of a and y, i.e.. y'= f(ax+by+c). To solve this problem we use the substitution var+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 we obtain an implicit solution x+ 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 y =
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