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Differential Equations - Degree of a differential equation

Grade 12CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The degree of a differential equation is the power (exponent) of the highest order derivative present in the equation, provided the differential equation is a polynomial in its derivatives (i.e., derivatives dydx,d2ydx2,…\frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots are not trapped inside functions like sin⁡,cos⁡,e,log⁡\sin, \cos, e, \log etc.).

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A differential equation must be expressed as a polynomial in terms of its derivatives to define its degree. If an equation like sin⁡(dydx)=x\sin\left(\frac{dy}{dx}\right) = x cannot be simplified to remove the derivative from the function argument, the degree is not defined.

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Before determining the degree, the differential equation must be made free from radicals (roots) and fractional powers with respect to the derivatives.

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While the order of a differential equation is always defined, the degree may or may not be defined.

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The degree is always a positive integer if it exists.

📐Formulae

P0(x,y)(dnydxn)k+P1(x,y)(dnydxn)k−1+⋯+Pk(x,y)=0P_0(x, y) \left( \frac{d^n y}{dx^n} \right)^k + P_1(x, y) \left( \frac{d^n y}{dx^n} \right)^{k-1} + \dots + P_k(x, y) = 0

Degree=Power of the highest order derivative (dnydxn)\text{Degree} = \text{Power of the highest order derivative } \left( \frac{d^n y}{dx^n} \right)

💡Examples

Problem 1:

Find the order and degree of the differential equation: (d2ydx2)3+(dydx)2+sin⁡(dydx)+1=0\left( \frac{d^2 y}{dx^2} \right)^3 + \left( \frac{dy}{dx} \right)^2 + \sin\left( \frac{dy}{dx} \right) + 1 = 0

Solution:

Order = 22, Degree = Not defined.

Explanation:

The highest order derivative is d2ydx2\frac{d^2 y}{dx^2}, so the order is 22. However, because the equation contains the term sin⁡(dydx)\sin\left( \frac{dy}{dx} \right), it is not a polynomial equation in its derivatives. Therefore, the degree is not defined.

Problem 2:

Find the degree of the differential equation: d2ydx2=1+(dydx)2\frac{d^2 y}{dx^2} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}

Solution:

Degree = 22.

Explanation:

To find the degree, we must remove the radical. Squaring both sides, we get: (d2ydx2)2=1+(dydx)2\left( \frac{d^2 y}{dx^2} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2. The highest order derivative is d2ydx2\frac{d^2 y}{dx^2} and its power is 22.

Problem 3:

Determine the order and degree of: y′′′+2(y′′)2+y′=0y''' + 2(y'')^2 + y' = 0

Solution:

Order = 33, Degree = 11.

Explanation:

The highest order derivative is y′′′y''' (which is d3ydx3\frac{d^3 y}{dx^3}), making the order 33. The power of this highest order derivative is 11, so the degree is 11.