The exponent of the highest derivative of a function in a non-radical and non-fractional differential equation is called the degree of a differential equation.

In differential equations, one or more derivatives of a function are involved as coefficients in the terms and the power of the highest derivative of the function is considered as the degree of the differential equations.

$4\dfrac{d^2y}{dx^2}-y+9 = 0$

It is a differential equation, where $y$ represents a function in $x$. In this case, the function $y$ is differentiated twice. So, we have to evaluate the exponent of the highest derivative of the function.

$\implies$ $4\Bigg(\dfrac{d^2y}{dx^2}\Bigg)^1-y+9 = 0$

Therefore, the degree of this differential equation is one and it is called the differential equation of first degree.

In some cases, the differential equations may contain fractions and radicals. So, the radicals and fractions should be eliminated from the differential equations firstly for evaluating the degree of any differential equation.

Let’s learn the concept of degree of differential equations from some more understandable examples.

$(1)\,\,\,$ $\dfrac{du}{d\theta} \,=\, 8u+7$

In this example, there is only one derivative and we have to evaluate its index to find the degree of differential equation.

$\implies$ $\Bigg(\dfrac{du}{d\theta}\Bigg)^1 \,=\, 8u+7$

Therefore, the degree of this differential equation is $1$ and it is called the differential equation of first degree.

$(2)\,\,\,$ $7\Bigg(\dfrac{d^2l}{dx^2}\Bigg)^{\Large \frac{5}{2}}+8\dfrac{dl}{dx}+9 = 0$

The variable $l$ represents a function in $x$ in the given differential equation. The function $l$ is differentiated two times in the first term and one time in the second term. The degree of the differential equation is defined by considering highest derivative but its exponent is a fraction. In fact, a degree of an equation cannot be a fraction. By eliminating the fraction form, the differential equation can be written as follows.

$\implies$ $49\Bigg(\dfrac{d^2l}{dx^2}\Bigg)^5$ $-$ $64\Bigg(\dfrac{dl}{dx}\Bigg)^2$ $-$ $144\dfrac{dl}{dx}$ $-$ $81$ $\,=\,$ $0$

In this example, the second derivative is highest derivative and its exponent is $5$. So, the degree of this differential equation is $5$. Hence, it is called the differential equation of fifth degree.

$(3)\,\,\,$ $\sqrt[\displaystyle 3]{\dfrac{d^9z}{dy^9}+5\dfrac{d^4z}{dy^4}-2\dfrac{dz}{dy}+16}$ $\,=\,$ $3$

In this example, the differential equation is in radical form. It is difficult for beginners to evaluate the degree of the differential equation. So, eliminate the root form and it is written as follows.

$\implies$ $\dfrac{d^9z}{dy^9}+5\dfrac{d^4z}{dy^4}-2\dfrac{dz}{dy}-11$ $\,=\,$ $0$

The ninth derivative is highest derivative in this case and its exponent is $1$. Therefore, the degree of this equation is one and it is called as the differential equation of first degree.

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