Question

find the order and degree if defined of the differential equation
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Answer 1

Step-by-step explanation:

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Answer 2

Answer:

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Step-by-step explanation:

Order of Differential Equation:-

Differential Equations are classified on the basis of the order. Order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.

For Example (i): d3xdx3+3xdydx=ey

In this equation the order of the highest derivative is 3 hence this is a third order differential equation.

Example (ii) : –(d2ydx2)4+dydx=3

This equation represents a second order differential equation.

This way we can have higher order differential equations i.e. nth order differential equations.

Differential-Equation

First order differential equation:

The order of highest derivative in case of first order differential equations is 1. A linear differential equation has order 1. In case of linear differential equations, the first derivative is the highest order derivative.

dydx+Py=Q

P and Q are either constants or functions of the independent variable only.

This represents a linear differential equation whose order is 1.

Example: dydx+(x2+5)y=x5

This also represents a First order Differential Equation.

Second Order Differential Equation:

When the order of the highest derivative present is 2, then it represents a second order differential equation.

Example: d2ydx2+(x3+3x)y=9

In this example, the order of the highest derivative is 2. Therefore, it is a second order differential equation.

Degree of Differential Equation:

The degree of differential equation is represented by the power of the highest order derivative in the given differential equation.

The differential equation must be a polynomial equation in derivatives for the degree to be defined.

Example 1:- d4ydx4+(d2ydx2)2–3dydx+y=9

Here, the exponent of the highest order derivative is one and the given differential equation is a polynomial equation in derivatives. Hence, the degree of this equation is 1.

Example 2: [d2ydx2+(dydx)2]4=k2(d3ydx3)2

The order of this equation is 3 and the degree is 2 as the highest derivative is of order 3 and the exponent raised to highest derivative is 2.

Example 3:- d2ydx2+cosd2ydx2=5x

The given differential equation is not a polynomial equation in derivatives. Hence, the degree for this equation is not defined.

Example 4:- (d3ydx3)2+y=0

The order of this equation is 3 and the degree is 2.

Example 5:- Figure out the order and degree of differential equation that can be formed from the equation 1–x2−−−−√+1–y2−−−−√=k(x–y).

Solution:-

Let x=sinθ,y=sinϕ

So, the given equation can be rewritten as

1–sinθ2−−−−−−−√+1–sinϕ2−−−−−−−√=k(sinθ–sinϕ)

⇒(cosθ+cosϕ)=k(sinθ–sinϕ)

⇒2cosθ+ϕ2cosθ–ϕ2=2kcosθ+ϕ2sinθ–ϕ2

cotθ–ϕ2=k

θ–ϕ=2cot−1k

sin−1x–sin−1y=2cot−1k

Differentiating both sides w. r. t. x, we get

11–x2–11–y2dydx=0

So, the degree of the differential equation is 1 and it is a first order differential equation.

In the upcoming discussions, we will learn about solutions to the various forms of differential equations