Order and degree
A doctor describes an illness by how serious it is and how it behaves. In the same way, a differential equation is described by its order (the highest derivative) and its degree (the power of that derivative).
The order is the order of the highest derivative in the equation. The degree is the power of this highest derivative, after the equation is made free of roots and fractions of the derivatives. If the equation cannot be written as a polynomial in the derivatives, the degree is not defined.
Rule 1: Find the order first — Look for the highest derivative.
- Easy: dy/dx + y = 0 has order 1 and degree 1.
- Medium: (d²y/dx²)³ + dy/dx = 0 has order 2 and degree 3.
- Exam-type: (d³y/dx³)² + (d²y/dx²)³ = 0 has order 3 and degree 2.
Rule 2: When the degree is not defined — Derivatives inside sin, cos, log or an exponential.
- Easy: dy/dx + sin(dy/dx) = 0 has order 1, and the degree is not defined.
- Medium: e^(dy/dx) = x has order 1, and the degree is not defined.
- Exam-type: (d²y/dx²)² = 1 + dy/dx, after squaring, has order 2 and degree 2.
Order is a count of the primes (the highest derivative). Degree is the power on that same highest derivative.
Taking the power of dy/dx as the degree when the highest derivative is d²y/dx².
Order is the highest tower, degree is its power.
Practice questions
- 1Write the order and degree of dy/dx = x².
- 2Write the order and degree of d²y/dx² + 3(dy/dx)² = 0.
- 3Write the order and degree of (d³y/dx³)² + (d²y/dx²)³ = 0.
- 4Write the order and degree of (dy/dx)³ + y = 0.
- 5Write the order and degree of dy/dx + sin(dy/dx) = 0.
- 6Write the order and degree of e^(dy/dx) = x.
- 7Write the order and degree of d²y/dx² = √(1 + dy/dx).
- 8Write the order and degree of (dy/dx)² = 1 + (d²y/dx²)².
Show answers (try them yourself first)
- 1order 1, degree 1
- 2order 2, degree 1
- 3order 3, degree 2
- 4order 1, degree 3
- 5order 1, degree not defined
- 6order 1, degree not defined
- 7Squaring gives (d²y/dx²)² = 1 + dy/dx, so order 2, degree 2
- 8order 2, degree 2
General and particular solutions
A dress pattern fits people of many sizes, but a tailor picks one size for you. The general solution is the pattern with a free constant, and the particular solution is the one made with your measurement.
A solution of a differential equation is a function that satisfies it. The general solution contains as many arbitrary constants as the order of the equation. When a condition is given, such as y = 4 at x = 1, the constants get fixed and the result is a particular solution.
Rule 1: General solution — The number of constants equals the order.
- Easy: dy/dx = 3x² has the general solution y = x³ + C.
- Medium: y″ + y = 0 has the general solution y = A cos x + B sin x with two constants.
- Exam-type: x dy/dx = y has the general solution y = Cx.
Rule 2: Particular solution — Put in the given point and find the constant.
- Easy: For y = x² + C passing through (1, 4), C = 3, so y = x² + 3.
- Medium: For y = Cx with y = 3 at x = 1, C = 3 and y = 3x.
- Exam-type: For y = A e^(2x) with y(0) = 5, A = 5 and y = 5 e^(2x).
To check whether a function is a solution, differentiate it and substitute in the equation.
Adding a constant to a particular solution, or giving too few constants for a higher-order equation.
Order tells how many constants.
Practice questions
- 1How many arbitrary constants does the general solution of a third-order equation contain?
- 2Show that y = x² is a solution of x dy/dx = 2y.
- 3Write the general solution of dy/dx = 3x².
- 4Find the particular solution of dy/dx = 3x² through (0, 1).
- 5Is y = eˣ a solution of d²y/dx² − y = 0?
- 6Is y = sin x a solution of d²y/dx² + y = 0?
- 7For y = x² + C find the particular solution with y(2) = 7.
- 8For dy/dx = 2y with general solution y = A e^(2x), find the solution with y(0) = 5.
Show answers (try them yourself first)
- 13
- 2x · 2x = 2x² = 2y, so the equation is satisfied
- 3y = x³ + C
- 4y = x³ + C and 1 = 0 + C give y = x³ + 1
- 5Yes, because d²y/dx² = eˣ = y
- 6Yes, because d²y/dx² = −sin x, so d²y/dx² + y = 0
- 7C = 7 − 4 = 3, so y = x² + 3
- 8y = 5 e^(2x)