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Degree OfDifferentialEquations

Definition Degree OfDifferentialEquations :

Let \(F\left ( x,y,{y}',{y}''\cdots y^{(n)} \right )= 0\)
be a differential equation of order n. If the given differential equation is a polynomial in \(y^{n}\) (nth derivative), then the highest degree of \( y^{n}\) is defined as the degree of the differential equation.

Note:

1. If the given equation nth derivative \(y^{n}\) enters in the denominator or has a fractional index, then it may be possible to free it from radicals by algebraic operations so that \(y^{n}\) has the least positive integral index and the equation is written as polynomial in \(y^{n}\).


2. The above definition of degree does not require variables x, t, u etc to be free from radicals and fractions.


3. If it is not possible to express the differential equation as a polynomial in \(y^{n}\) , then the degree of the differential equation is not defined.


Examples:

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

\(\Rightarrow \left ( y-x\frac{dy}{dx} \right )^{2}=1+\left ( \frac{dy}{dx} \right )^{2}\)

\(\Rightarrow \left ( 1+x^{2} \right )\left ( \frac{dy}{dx} \right )^{2}+2xy\frac{dy}{dx}+\left ( 1-y^{2} \right )=0\)

This is a polynomial equation in \(\frac{dy}{dx}\). The highest degree of \(\frac{dy}{dx}\) is 2.

Hence the degree of the above differential equation is 2.

2 Example :

 \( a \frac{\mathrm{d^{2}}y }{\mathrm{d} x^{2}}\)=\(\left [ 1 + \left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2} \right ]^{\frac{3}{2}}\)

Sol) Squaring on both sides we get ,

\(a^{2}  \left ( \frac{\mathrm{d^{2}}y }{\mathrm{d} x^{2}} \right )^{2}\)=\(\left [ 1 + \left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2} \right ]^{3}\)

This is a polynomial equation in \(\frac{\mathrm{d^{2}}y }{\mathrm{d} x^{2}}\) . The highest degree of \(\frac{\mathrm{d^{2}}y }{\mathrm{d} x^{2}}\) is 2. Hence the degree of the differential equation is 2 .

3 Example :

y = cos(\(\frac{dy}{dx}\)) and y = x + ln(\(\frac{dy}{dx}\))

Sol) Both the equations cannot be expressed as polynomial equations in \(\frac{dy}{dx}\) .

Hence the degree of the above differential equations cannot be determined and hence undefined .

This is explanation with examples on Degree OfDifferentialEquations .



Differential Equations


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