The equation p sin α = constant is known as which equation?

The equation p sin α = constant is known as which equation?

• A. Bernoulli's equation
• B. Clairaut's equation
• C. Laplacian equation
• D. Navier equation

Answer: (B)

Clairaut's equation involves expressing the relationship between y, x, and the slope p = dy/dx in a way that y depends linearly on x through p, as in y = x p + f(p). In the plane, the tangent angle α satisfies p = tan α, and sin α is p divided by the length of the slope vector, giving sin α = p / sqrt(1+p^2). The condition p sin α = constant then becomes a relation that depends only on p (and not directly on x or y): p^2 / sqrt(1+p^2) = constant. This kind of slope-only constraint is a geometric signature of Clairaut-type problems, where the solution set consists of a family of straight lines with slope p and an envelope formed by the parameter p. In other words, the equation encodes the same structure as y = x p + f(p), with p = dy/dx, whose general solution is a family of lines and whose singular solution is the envelope. That connection is why this is identified as Clairaut's equation.

Clairaut's equation involves expressing the relationship between y, x, and the slope p = dy/dx in a way that y depends linearly on x through p, as in y = x p + f(p). In the plane, the tangent angle α satisfies p = tan α, and sin α is p divided by the length of the slope vector, giving sin α = p / sqrt(1+p^2). The condition p sin α = constant then becomes a relation that depends only on p (and not directly on x or y): p^2 / sqrt(1+p^2) = constant. This kind of slope-only constraint is a geometric signature of Clairaut-type problems, where the solution set consists of a family of straight lines with slope p and an envelope formed by the parameter p. In other words, the equation encodes the same structure as y = x p + f(p), with p = dy/dx, whose general solution is a family of lines and whose singular solution is the envelope. That connection is why this is identified as Clairaut's equation.