Q5. Find the order and degree of y″ + (y′)2 + 2y = 0 (NCERT Exercise 9.1, Q9, page 304).
Order 2 and degree 1: the square is on y′, not on the highest derivative y″.
Q6. Find the order and degree, if defined, of xy d2y/dx2 + x(dy/dx)2 − y dy/dx = 0 (NCERT Example 1 (ii), page 303).
The highest derivative is the second, to the power 1: order 2, degree 1.
Q7. Beyond the NCERT text (JEE): find the order and degree of (1 + (y′)2)3/2 = 4y″.
As written it is not a polynomial in y′, because of the power 3/2. Squaring both sides gives (1 + (y′)2)3 = 16(y″)2, a polynomial in the derivatives. So the order is 2 and the degree is 2. This is the usual degree after clearing radicals. The original equation also requires y″ > 0, because its left side is positive; the squared equation alone also admits an unwanted negative branch.
Q8. Find the order and degree of (d3y/dx3)2 − 5x(dy/dx)4 + ex = 0.
The highest derivative is the third, squared: order 3, degree 2. The fourth power of the first derivative does not decide the degree, and ex is a function of x.
Q9. Find the order and degree of y″ + log (y′) = 0.
Order 2. The degree is not defined, because log (y′) is not a polynomial in y′.
Q10. Write a differential equation of order 2 and degree 3, and one of order 3 whose degree is not defined.
For example, (y″)3 + y′ = x, of order 2 and degree 3; and y‴ + sin (y″) = 0, of order 3 with no degree. Many other answers are correct.