To find the differential equation for the family of tangents to the curve y=x^2, one must use the derivative, which is y'=2a, to establish the equation of the tangent line at any point a. The equation of the tangent line can be expressed as y - y(a) = 2a(x - a). For the normals, which are perpendicular to the tangents, the slope is -1/(2a), leading to the normal line equation y - y(a) = -1/(2a)(x - a). Both the tangents and normals can be represented by their respective differential equations derived from these relationships. Understanding these concepts is crucial for solving problems related to tangents and normals in calculus.