Form the differential equation of the family of straight line with slope and y- intercept equal
To find the differential equation of a family of straight lines with equal slope and y-intercept, let's assume the slope and y-intercept are represented by the parameters m and c, respectively.
The equation of a straight line is given by:
y = mx + c
Since we want the lines to have equal slope and y-intercept, we can write the equation as:
y = mx + c1, where c1 is a constant
Differentiating both sides of the equation with respect to x, we get:
dy/dx = d/dx(mx + c1)
The derivative of c1 with respect to x is zero because it is a constant. The derivative of mx with respect to x is simply m. Thus, we have:
dy/dx = m
Therefore, the differential equation of the family of straight lines with equal slope and y-intercept is:
dy/dx = m, where m is the constant slope.
To form the differential equation of a family of straight lines with equal slope and y-intercept, let's start by considering the general form of a straight line equation:
y = mx + c
where m is the slope and c is the y-intercept. Since we want the slope and y-intercept to be equal, we can write the equation as:
y = m*x + m
Now, to form the differential equation, we need to find the relation between the variables x and y. We can do this by taking the derivative of both sides with respect to x.
d(y) / d(x) = d(m*x + m) / d(x)
d(y) / d(x) = m
The left side of the equation represents the derivative of y with respect to x, which is dy/dx. So, our differential equation becomes:
dy/dx = m
Since m represents the common slope and is a constant value, this differential equation represents a family of straight lines with a fixed slope.
Straight line in slope intercept form:
y = m x + b
Differentiate:
dy = m dx + 0
dy = m dx
dy / dx = m