find the differential equation of family of lines with slope and y intercept equal.
To find the differential equation of a family of lines with equal slope and y-intercept, we need to find the relationship between the slope and the y-intercept.
Let's denote the slope by 'm' and the y-intercept by 'b'. Since all lines in this family have the same slope and y-intercept, the equation of a generic line in this family can be written as:
y = mx + b
To find the differential equation, we need to eliminate the constant term 'b'. Since the slope 'm' is equal to the y-intercept 'b', we can substitute 'b' with 'm' in the equation:
y = mx + m
Simplifying it, we have:
y = m(x + 1)
Now, to find the differential equation, we differentiate both sides with respect to 'x'. Using the power rule of differentiation, we get:
dy/dx = m
Hence, the differential equation for the family of lines with equal slope and y-intercept is:
dy/dx = m
where 'm' represents the common slope of all lines in this family.