Basic Concepts: Order and Degree

Introduction to Differential Equations

An equation involving one or more derivatives of a dependent variable with respect to an independent variable is called a differential equation.

For example, in $$\frac{dy}{dx} + 5y = e^x,$$

• $x$ is the independent variable,
• $y$ is the dependent variable,
• $\frac{dy}{dx}$ is the derivative of the dependent variable with respect to the independent variable.

Differential equations are broadly classified into two types:

1. Ordinary Differential Equations (ODEs)

These involve derivatives with respect to only one independent variable, such as $$\frac{dy}{dx} + y = 0.$$

2. Partial Differential Equations (PDEs)

These involve partial derivatives with respect to more than one independent variable.

This chapter deals only with ordinary differential equations.

Order of a Differential Equation

The order of a differential equation is the order of the highest order derivative present in the equation.

To determine the order, identify the derivative that has been differentiated the maximum number of times.

Examples:

• In $$\frac{dy}{dx} = \cos x,$$ the highest derivative is the first derivative. So, the order is 1.

• In $$\frac{d^2y}{dx^2} + y = 0,$$ the highest derivative is the second derivative. So, the order is 2.

• In $$\left(\frac{d^3y}{dx^3}\right)^2 + x^2\left(\frac{d^2y}{dx^2}\right)^4 = 0,$$ the highest derivative is $\frac{d^3y}{dx^3}$. So, the order is 3.

Important:

• Order depends only on which derivative is highest, not on its power.
• The order of a differential equation is always a positive integer.

Degree of a Differential Equation

The degree of a differential equation is the power of the highest order derivative, provided the equation can be expressed as a polynomial in its derivatives.

This condition is extremely important.

A differential equation is said to be polynomial in derivatives if:

• derivatives appear only with non-negative integer powers,
• derivatives are not inside trigonometric, logarithmic, exponential, or other transcendental functions,
• and there are no fractional powers of derivatives left after simplification.

Steps to find the degree

1. Find the order, i.e. identify the highest order derivative.
2. Rewrite the equation, if needed, so that it becomes a polynomial in derivatives.
3. Then look at the exponent of the highest order derivative. That exponent is the degree.

Example: $$\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^4 + y = 0$$ Here the highest order derivative is $\frac{d^2y}{dx^2}$, and its power is 3. So,

• Order = 2
• Degree = 3

When is the Degree Not Defined?

The degree of a differential equation is not defined if the equation cannot be reduced to a polynomial form in its derivatives.

This happens when a derivative appears inside a transcendental or non-algebraic function and cannot be algebraically freed in polynomial form.

Examples where degree is not defined:

• $$\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$$ because $\frac{dy}{dx}$ is inside a sine function.

• $$e^{\frac{dy}{dx}} + y = x$$ because the derivative is in the exponent.

• $$\log\left(\frac{d^2y}{dx^2}\right) = x$$ if left in this form, the derivative is inside a logarithm. However, if it can be algebraically isolated and rewritten polynomially, then degree may become defined.

Very important caution

In $$\frac{dy}{dx} = \sin(x+y),$$ the sine function contains only $x$ and $y$, not a derivative. So the equation is polynomial in the derivative $\frac{dy}{dx}$, and therefore the degree is 1.

Example 1: Basic Linear Equation

Find the order and degree of the differential equation: $x \frac{dy}{dx} + y = 0$.

Solution: Step 1: Identify the highest order derivative. The only derivative present is $\frac{dy}{dx}$, which is a first-order derivative. Therefore, the order is 1.

Step 2: Check whether the equation is polynomial in derivatives. Yes. The derivative $\frac{dy}{dx}$ appears linearly and is not inside any function.

Step 3: Determine the degree. The highest order derivative is $\frac{dy}{dx}$ and its power is 1. Therefore, the degree is 1.

Answer: Order = 1, Degree = 1.

Example 2: Higher Order

Find the order and degree of the differential equation: $\frac{d^2y}{dx^2} + 5x \left(\frac{dy}{dx}\right)^2 = 6y$.

Solution: Step 1: Identify the highest order derivative. The derivatives present are $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. The highest among these is $\frac{d^2y}{dx^2}$. Hence, the order is 2.

Step 2: Check whether it is polynomial in derivatives. Yes. Both derivatives occur with non-negative integer powers, and neither is inside a transcendental function.

Step 3: Determine the degree. The highest order derivative is $\frac{d^2y}{dx^2}$ and its exponent is 1. Therefore, the degree is 1.

Answer: Order = 2, Degree = 1.

Example 3: Higher Degree

Find the order and degree of the differential equation: $\left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^5 + y = 0$.

Solution: Step 1: Identify the highest order derivative. The derivatives present are second and third derivatives. The highest order derivative is $\frac{d^3y}{dx^3}$. Hence, the order is 3.

Step 2: Check the polynomial condition. The equation is polynomial in derivatives because all derivative terms have non-negative integer powers.

Step 3: Determine the degree. The highest order derivative is $\frac{d^3y}{dx^3}$, and it is raised to the power 2. Therefore, the degree is 2.

Answer: Order = 3, Degree = 2.

Example 4: Degree Not Defined (Trigonometric)

Find the order and degree of the differential equation: $\frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0$.

Solution: Step 1: Identify the highest order derivative. The highest order derivative is $\frac{d^2y}{dx^2}$. Hence, the order is 2.

Step 2: Check whether the equation is polynomial in derivatives. No. The derivative $\frac{dy}{dx}$ is inside the cosine function. Therefore, the equation is not polynomial in derivatives.

Step 3: Conclude the degree. Since the polynomial condition fails, the degree is not defined.

Answer: Order = 2, Degree = Not defined.

Example 5: Clearing Fractional Powers

Find the order and degree of the differential equation: $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = 5 \frac{d^2y}{dx^2}$.

Solution: Step 1: Identify the highest order derivative. The highest order derivative present is $\frac{d^2y}{dx^2}$. So, the order is 2.

Step 2: Convert the equation into polynomial form in derivatives. The left side contains a fractional exponent $3/2$. To remove it, square both sides: $$\left(\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3/2}\right)^2 = \left(5\frac{d^2y}{dx^2}\right)^2$$ which gives $$\left[1+\left(\frac{dy}{dx}\right)^2\right]^3 = 25\left(\frac{d^2y}{dx^2}\right)^2.$$ Now the equation is polynomial in derivatives.

Step 3: Determine the degree. The highest order derivative is $\frac{d^2y}{dx^2}$ and its power is now 2. Therefore, the degree is 2.

Answer: Order = 2, Degree = 2.

Example 6: Clearing Radicals

Find the order and degree of the differential equation: $y = x \frac{dy}{dx} + a \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$.

Solution: Step 1: Identify the highest order derivative. Only $\frac{dy}{dx}$ appears, so the order is 1.

Step 2: Remove the radical. First isolate the square root term: $$y - x\frac{dy}{dx} = a\sqrt{1+\left(\frac{dy}{dx}\right)^2}.$$ Now square both sides: $$\left(y - x\frac{dy}{dx}\right)^2 = a^2\left(1+\left(\frac{dy}{dx}\right)^2\right).$$ Expanding and rearranging: $$y^2 - 2xy\frac{dy}{dx} + x^2\left(\frac{dy}{dx}\right)^2 = a^2 + a^2\left(\frac{dy}{dx}\right)^2,$$ so $$(x^2-a^2)\left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} + (y^2-a^2)=0.$$ This is now polynomial in the derivative.

Step 3: Determine the degree. The highest order derivative is still $\frac{dy}{dx}$, and its highest power is 2. Therefore, the degree is 2.

Answer: Order = 1, Degree = 2.

Example 7: Degree Not Defined (Exponential)

Find the order and degree of the differential equation: $y''' + y^2 + e^{y'} = 0$.

Solution: Step 1: Identify the highest order derivative. The highest derivative present is $y'''$, i.e. $\frac{d^3y}{dx^3}$. So, the order is 3.

Step 2: Check the polynomial condition. The first derivative $y'$ appears inside an exponential function. Hence the equation is not polynomial in derivatives.

Step 3: Conclude the degree. Since it is not polynomial in derivatives, the degree is not defined.

Answer: Order = 3, Degree = Not defined.

Prime notation is often used for derivatives: $y' = \frac{dy}{dx}$, $y'' = \frac{d^2y}{dx^2}$, $y''' = \frac{d^3y}{dx^3}$.

Example 8: Resolving Inverse Trigonometric Functions

Find the order and degree of the differential equation: $\sin^{-1}\left(\frac{dy}{dx}\right) = x + y$.

Solution: Step 1: At first glance, the derivative is inside an inverse trigonometric function, so the degree is not immediately visible.

Step 2: Isolate the derivative algebraically. Take sine on both sides: $$\frac{dy}{dx} = \sin(x+y).$$ Now the derivative is no longer inside a transcendental function.

Step 3: Determine the order. The highest order derivative is $\frac{dy}{dx}$, so the order is 1.

Step 4: Determine the degree. Now the equation is polynomial in the derivative $\frac{dy}{dx}$, and its power is 1. Therefore, the degree is 1.

Answer: Order = 1, Degree = 1.

Example 9: Nested Derivatives

Find the order and degree of the differential equation: $\frac{d}{dx}\left( y \frac{dy}{dx} \right) = 0$.

Solution: Step 1: Expand the derivative using the product rule. $$\frac{d}{dx}\left(y\frac{dy}{dx}\right) = y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2.$$ So the equation becomes $$y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0.$$

Step 2: Determine the order. The highest order derivative is $\frac{d^2y}{dx^2}$. Therefore, the order is 2.

Step 3: Determine the degree. The equation is polynomial in derivatives, and the highest order derivative $\frac{d^2y}{dx^2}$ has power 1. Therefore, the degree is 1.

Answer: Order = 2, Degree = 1.

Example 10: Multiple Fractional Powers

Find the order and degree of the differential equation: $\left(\frac{d^2y}{dx^2}\right)^{2/3} = \left(\frac{dy}{dx}\right)^{1/2}$.

Solution: Step 1: Identify the highest order derivative. The highest order derivative is $\frac{d^2y}{dx^2}$. Thus, the order is 2.

Step 2: Remove fractional exponents. The powers are $2/3$ and $1/2$. The LCM of denominators 3 and 2 is 6. Raise both sides to the 6th power: $$\left[\left(\frac{d^2y}{dx^2}\right)^{2/3}\right]^6 = \left[\left(\frac{dy}{dx}\right)^{1/2}\right]^6.$$ This gives $$\left(\frac{d^2y}{dx^2}\right)^4 = \left(\frac{dy}{dx}\right)^3.$$ Now the equation is polynomial in derivatives.

Step 3: Determine the degree. The highest order derivative is $\frac{d^2y}{dx^2}$, and its exponent is 4. Therefore, the degree is 4.

Answer: Order = 2, Degree = 4.