Find the order and degree (if defined) of: (i) (d²y/dx²)³ + x(dy/dx)⁴ = y, (ii) d³y/dx³ + sin(d²y/dx²) = 0, (iii) d⁴y/dx⁴ − 3(d²y/dx²)⁵ + y = 0.

1. 1 · Write what is given and what is to be found. Three DEs. To find: the order and degree of each, with a reason.
2. 2 · Write the rule. Order = order of the highest derivative. Degree = its power, once the DE is a polynomial in its derivatives.
3. 3 · Part (i).
   • Highest derivative: d²y/dx², so the order is 2.
   • It is a polynomial in the derivatives; d²y/dx² has power 3, so the degree is 3.
4. 4 · Part (ii).
   • Highest derivative: d³y/dx³, so the order is 3.
   • d²y/dx² is inside sin, so it is not a polynomial in derivatives: degree not defined.
5. 5 · Part (iii).
   • Highest derivative: d⁴y/dx⁴, so the order is 4.
   • It is a polynomial in the derivatives. d⁴y/dx⁴ has power 1, so the degree is 1. The power 5 belongs to a lower derivative.
6. 6 · Check and box the answer. (i) order 2, degree 3; (ii) order 3, degree not defined; (iii) order 4, degree 1.

Scoring tip: When the degree is not defined, write the reason 'not a polynomial in its derivatives'. A bare 'not defined' can lose the mark.