Order, Degree and Solutions: Quick Revision

A brief note on Order, Degree and Solutions. Read it before a test. Try each question at the end before you open its answer.

Order and degree

• Differential equation: an equation with a function and its derivatives.
• Order: the order of the highest derivative.
• Degree: the power of the highest derivative, after clearing roots and fractions of derivatives.
• Not defined: the degree has no value when a derivative sits inside $\sin$, $\cos$, $e^{(\ )}$ or $\ln$.

Solutions

• General solution: has as many independent constants as the order.
• Particular solution: the constants are fixed by conditions, one for each constant.
• Check: put the function and its derivatives into the equation.

Forming an equation

• Count the independent constants, say $n$.
• Differentiate $n$ times and eliminate the constants.
• Example: $y = Cx^3$ gives $x\,y' = 3y$. Lines $y = mx$ give $x\,y' = y$.

Direction fields

• Slope: $\dfrac{dy}{dx} = f(x, y)$ gives a slope at every point.
• Curves: solution curves follow the slopes and never cross.
• Step: $y_{\text{new}} = y_{\text{old}} + h\,f(x_{\text{old}}, y_{\text{old}})$.

A worked example

Find the order and the degree of $\dfrac{d^2y}{dx^2} = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^3}$.

1. Clear the root. Square both sides: $\left(\dfrac{d^2y}{dx^2}\right)^2 = 1 + \left(\dfrac{dy}{dx}\right)^3$.
2. Read off. The highest derivative is the second, so the order is $2$. Its power is $2$, so the degree is $2$.

Practice with new numbers

A program to try

The program checks that $y = 2e^{3x}$ solves $y' = 3y$, and that $y = 5x^2$ solves $x\,y' = 2y$, at three values of $x$.

Mistakes to avoid

• Order and degree mixed up. The order counts derivatives. The degree is a power.
• A degree read before clearing roots. Clear them first.
• Too few constants. An equation of order $n$ has $n$ constants in the general solution.
• A constant left in the formed equation. Eliminate all of them.

Check yourself

1. State the order and the degree of $\left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = x$.

2. How many constants does the general solution of a third-order equation have?

3. Form the equation of the family $y = Cx^2$.

4. Is $y = e^{2x}$ a solution of $y'' - 4y = 0$?

5. For $\dfrac{dy}{dx} = x - y$ at $(0, 1)$, find the height after one step of $h = 0.5$.

Sources

• OpenStax, Calculus Volume 2, Chapter 4, sections 4.1 and 4.2.
• NCERT, Mathematics Part II for Class XII, Chapter 9, the sections on basic concepts and on formation of a differential equation.