The order and degree of differential equation of all tangent lines to the parabola x^2=4 y is respectively.

Step-by-step solution

Differential equation of tangent lines to the parabola x^2 = 4y . The general equation of a tangent line to x^2 = 4y with slope m is y = mx - m^2 . Differentiating with respect to x gives dy dx = m . Substituting m = dy dx into the tangent equation yields y = x ( dy dx ) - ( dy dx )^2 . Rewriting the differential equation as ( dy dx )^2 - x ( dy dx ) + y = 0 shows the highest derivative is dy dx , which is first order. The power of this derivative is 2, so the degree is 2. The order is 1 and the degree is 2.