Question Statement

Find the order and degree of each of the following differential equations:

(i) $(1-x)y^{\prime \prime }-4x{y}^{\prime }+5y=\cos x$

(ii) $y{y}^{\prime }+2y=1+x^2$

(iii) ${\left(y^{\prime \prime }\right)}^3-3y^{\prime }+2y=x$

(iv) ${\left(y^{\prime }\right)}^2-y{y}^{\prime \prime }+2=0$

(v) $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3=0$

(vi) $\frac{dy}{dx}=\sqrt{1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}$

(vii) $x^3\frac{d^{4}y}{d{x}^4}-x^2\frac{d^{2}y}{d{x}^2}+4{\left(\frac{dy}{dx}\right)}^5+4xy-3y=0$

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Background and Explanation

The order of a differential equation is the order of the highest derivative present in the equation. The degree is the power of the highest order derivative, provided the differential equation is expressed as a polynomial in its derivatives (meaning no fractional powers or radicals involving the derivatives).

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Solution

Part (i)

Equation: $(1-x)y^{\prime \prime }-4x{y}^{\prime }+5y=\cos x$ The highest derivative present is $y^{\prime \prime }$ (the second derivative). The power of $y^{\prime \prime }$ is 1.

• Order $=2$
• Degree $=1$

Part (ii)

Equation: $y{y}^{\prime }+2y=1+x^2$ The highest derivative present is $y^{\prime }$ (the first derivative). The power of $y^{\prime }$ is 1.

• Order $=1$
• Degree $=1$

Part (iii)

Equation: ${\left(y^{\prime \prime }\right)}^3-3y^{\prime }+2y=x$ The highest derivative present is $y^{\prime \prime }$ (the second derivative). The power of this highest derivative is 3.

• Order $=2$
• Degree $=3$

Part (iv)

Equation: ${\left(y^{\prime }\right)}^2-y{y}^{\prime \prime }+2=0$ The highest derivative present is $y^{\prime \prime }$ (the second derivative). Even though $y^{\prime }$ is squared, we only look at the power of the highest order derivative, which is 1.

• Order $=2$
• Degree $=1$

Part (v)

Equation: $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3=0$ The highest derivative present is $\frac{d^{2}y}{d{x}^2}$ (the second derivative). Its power is 1.

• Order $=2$
• Degree $=1$

Part (vi)

Equation: $\frac{dy}{dx}=\sqrt{1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}$ To find the degree, we must first remove the radical to make the equation a polynomial in derivatives. Square both sides: ${\left(\frac{dy}{dx}\right)}^2=1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$ The highest derivative is $\frac{d^{2}y}{d{x}^2}$ (the second derivative). Its power in this form is 2.

• Order $=2$
• Degree $=2$

Part (vii)

Equation: $x^3\frac{d^{4}y}{d{x}^4}-x^2\frac{d^{2}y}{d{x}^2}+4{\left(\frac{dy}{dx}\right)}^5+4xy-3y=0$ The highest derivative present is $\frac{d^{4}y}{d{x}^4}$ (the fourth derivative). Its power is 1.

• Order $=4$
• Degree $=1$

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Key Formulas or Methods Used

• Order: Identified by the term $\frac{d^{n}y}{d{x}^n}$ where $n$ is the largest integer in the equation.
• Degree: The exponent of the highest order derivative after the equation is cleared of radicals and fractions involving derivatives.
• Polynomial Form: A differential equation must be a polynomial in terms of its derivatives ($y^{\prime },y^{\prime \prime },\dots$) to define a degree.

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Summary of Steps

1. Identify the highest order derivative present in the equation to determine the Order.
2. Check if the equation contains radicals or fractional powers involving the derivatives.
3. If radicals exist, rationalize the equation (e.g., by squaring) to express it as a polynomial in derivatives.
4. Identify the exponent of the highest order derivative to determine the Degree.