The difference between degree and order of a differential equation that represents the family of curves given by y² = a( x + √a/2 ), a > 0 is _________.

Solution

Maths sir · 4 steps

One constant means one differentiation, but clear the square root before you read the degree.

1. Find the order. The family has only one arbitrary constant, a, so the differential equation has order 1. Differentiate once: 2y y' = a.
2. Eliminate a. Put a = 2yy' back into the curve: y² = 2yy'(x + √2yy'/2). Divide by y and rearrange to get y - 2xy' = y'√2yy'.
3. Remove the radical. Degree is defined only once the equation is a polynomial in the derivatives, so square both sides.

   (y - 2xdy/dx)² = 2y(dy/dx)³

4. So. The highest power of dy/dx is 3, so the degree is 3. Degree minus order is 3 - 1 = 2.

The common slip: 1. Taking the degree as 2 from the 4x²(y')² term on the left and missing the (y')³ on the right gives 1.

The idea to keep. Number of arbitrary constants gives the order. For the degree, first make the equation free of radicals and fractional powers in the derivatives, then read the highest power of the highest derivative.

Answer from the NTA final answer key. Solution by PrepWiser.