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Differential Equations - Basic Concepts

Grade 12CBSE

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

🔑Concepts

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A differential equation is an equation involving derivatives of a dependent variable with respect to one or more independent variables. For example, xdydx+y=0x \frac{dy}{dx} + y = 0.

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The Order of a differential equation is the order of the highest order derivative appearing in the equation. It is always a positive integer.

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The Degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial equation in its derivatives. If the equation is not a polynomial in derivatives (e.g., involves sin⁡(dydx)\sin(\frac{dy}{dx}) or edydxe^{\frac{dy}{dx}}), the degree is not defined.

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A General Solution is a solution of a differential equation that contains as many arbitrary constants as the order of the equation.

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A Particular Solution is a solution obtained from the general solution by assigning specific values to the arbitrary constants, often based on given initial conditions.

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To form a differential equation from a given family of curves, differentiate the equation with respect to the independent variable as many times as the number of arbitrary constants, and then eliminate those constants.

📐Formulae

General form: F(x,y,dydx,d2ydx2,…,dnydxn)=0\text{General form: } F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots, \frac{d^ny}{dx^n}\right) = 0

Order=n where dnydxn is the highest derivative.\text{Order} = n \text{ where } \frac{d^ny}{dx^n} \text{ is the highest derivative.}

Degree=k in (dnydxn)k+⋯=0 (if polynomial in derivatives).\text{Degree} = k \text{ in } \left(\frac{d^ny}{dx^n}\right)^k + \dots = 0 \text{ (if polynomial in derivatives).}

💡Examples

Problem 1:

Find the order and degree (if defined) of the differential equation: (d2ydx2)3+(dydx)2+sin⁡(dydx)+1=0\left(\frac{d^2y}{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 present in the equation is d2ydx2\frac{d^2y}{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:

Verify that the function y=e−3xy = e^{-3x} is a solution of the differential equation d2ydx2+dydx−6y=0\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0

Solution:

Differentiating y=e−3xy = e^{-3x}, we get dydx=−3e−3x\frac{dy}{dx} = -3e^{-3x} and d2ydx2=9e−3x\frac{d^2y}{dx^2} = 9e^{-3x}. Substituting these into the LHS: 9e−3x+(−3e−3x)−6(e−3x)=(9−3−6)e−3x=09e^{-3x} + (-3e^{-3x}) - 6(e^{-3x}) = (9 - 3 - 6)e^{-3x} = 0. Since LHS = RHS, the function is a solution.

Explanation:

To verify a solution, compute the required derivatives of the given function and substitute them into the differential equation to see if the equation holds true.

Problem 3:

Find the order and degree of the differential equation: xyd2ydx2+x(dydx)2−ydydx=0xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0

Solution:

Order = 22, Degree = 11.

Explanation:

The highest order derivative is d2ydx2\frac{d^2y}{dx^2}, making the order 22. The power of this highest order derivative is 11, and since the equation is a polynomial in derivatives, the degree is 11.