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MATHMAA

# 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 .