Substitution Method

Substitution Method

Let the differential equation be of the form dy/dx = f (ax + by + c).

(i) If a ≠ 0 and b ≠ 0 , then the substitution ax + by + c = z reduces the given equation to the variables separable form.

(ii) If a = 0 or b = 0 , then the differential equation is already in separable form.

Example 10.16

Solution

To solve the given differential equation, we make the substitution 3x + y + 4 = z.

Differentiating with respect to x, we get dy/dx = dz/dx − 3 . So the given differential equation becomes 

In this equation variables are separable. So, separating the variables and integrating, we get the general solution of the given differential equation as