In case an equation is in the form y′=f(ax+by+c)y′=f(ax+by+c), i. e., the RHS is a linear function of xx and yy. We will use the substitution v=ax+by+cv=ax+by+c to find an implicit general solution. The right hand side of the following first order problem.

The right hand side Of the following first order problem
y' = (5x — 3y+ 1)^1/2 + 5/3, y(0)=0

is a function Of a linear combination Of z and y, i. e y'= f(ax+by+c). To solve this problem use the substitution v= ax + by +c which transforms the equation into a separable equation.

We obtain the following separable equation in the variables Z and V:

a. Solving this equation an implicit general solution in terms 0f x, v can be written in the form:

x+ = C

Transforming back to the variables Z 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=