Mathematics · Ch 13 — Differential Equations
Order and Degree of the Differential Equation
Order and Degree of the Differential Equation
Two basic labels classify every differential equation: its ORDER and its DEGREE.
The ORDER of a differential equation is the order of the highest derivative that appears in it. If the highest derivative present is dy/dx, the order is 1; if it is d²y/dx², the order is 2; and so on — the order is found simply by scanning the equation for the highest-numbered derivative, with no algebraic manipulation required first.
The DEGREE of a differential equation is the power to which that highest-order derivative is raised, but this can only be read off once the equation has been rewritten as a POLYNOMIAL equation in the derivatives — meaning every derivative present, of whatever order, occurs with a non-negative integer power, with no square roots, fractional exponents, or derivatives trapped inside transcendental functions such as sin(·), cos(·), e^(·) or log(·). Often the equation is not handed over in this clean polynomial form, and some algebra (typically squaring, cubing, or otherwise raising both sides to a power, or clearing a fraction by cross-multiplying) is needed first to remove a radical or a derivative sitting in a denominator. Once that is done, the degree is simply the exponent on the highest-order derivative term. If, even after every available algebraic step, some derivative (whether the highest-order one or a lower one) is still trapped inside a transcendental function, the equation can never become a polynomial in the derivatives, and the degree is declared NOT DEFINED — the order, however, is still perfectly well defined in that situation.
Nine solved examples work through this classification:
- x²(d²y/dx²) + 3x(dy/dx) + 4y = 0 — already polynomial; order 2, degree 1.
- (d³y/dx³)² + xy(dy/dx) − 2x + 3y + 7 = 0 — already polynomial; order 3, degree 2.
- r(dr/dθ) + cos θ = 5 — already polynomial (only one derivative, to the first power); order 1, degree 1.
- (d²y/dx²)² + (dy/dx)² = e^x — already polynomial; order 2, degree 2.
- dy/dx + 3xy/(dy/dx) = cos x — has dy/dx in a denominator; multiplying through by dy/dx clears it, giving (dy/dx)² + 3xy = cos x·(dy/dx); order 1, degree 2.
- √[1 + 1/(dy/dx)²] = (d²y/dx²)^(3/2) — squaring both sides to clear the square root and the 3/2 power together gives 1 + 1/(dy/dx)² = (d²y/dx²)³, and multiplying through by (dy/dx)² gives (dy/dx)² + 1 = (d²y/dx²)³·(dy/dx)²; order 2, degree 3.
- d⁴y/dx⁴ = [1 + (dy/dx)²]³ — already polynomial once the cube is expanded on the right (only the highest derivative d⁴y/dx⁴ is isolated on the left, to the first power); order 4, degree 1.
- e^(dy/dx) + dy/dx = x — the derivative dy/dx is trapped inside e^(dy/dx), a transcendental function, and no algebraic step removes it while keeping the equation polynomial; order 1, but the degree is NOT DEFINED. …
Worked out. The textbook records two standing notational facts right after the worked examples: dy/dx is also written y', d²y/dx² is also written y'', and d³y/dx³ is also written y''', with the pattern continuing for higher derivatives — this shorthand is used freely in later sections and exercises. The second note states the general rule that the order and the degree of a differential equation, whenever the degree is defined at all, are always positive integers — never zero, negative, or fractional — which is exactly the fact used to declare a degree 'not defined' whenever a derivative is trapped inside a transcendental function like sin, cos, e(·) or log(·), since no algebraic …