Question

For the following differential equation givne below indicate its order and degree (when defined)

$\frac{d^{2}y}{d{x}^2}+5x{\left(\frac{dy}{dx}\right)}^2-6y=logx$

${\left(\frac{dy}{dx}\right)}^3-4{\left(\frac{dy}{dx}\right)}^2+7y=sinx$

$\frac{d^{4}y}{d{x}^4}-sin\frac{d^{3}y}{d{x}^3}=0$

Answer 1

The given differential equations is
$\frac{d^{2}y}{d{x}^2}+5x{\left(\frac{dy}{dx}\right)}^2-6y=logx$
​​​​$\frac{d^{2}y}{d{x}^2}+5x{\left(\frac{dy}{dx}\right)}^2-6y=logx\Rightarrow \frac{d^{2}y}{d{x}^2}+5x{\left(\frac{dy}{dx}\right)}^2-6y-logx=0$
As the order of the highest order derivative that occurs in the given differential equation is $\frac{d^{2}y}{d{x}^2}.$ Thus, its order is 2. Also index of the highest order derivation $\frac{d^{2}y}{d{x}^2}$ is one. Hence, degree of the given differential equations is 1.

Note:In any differential equation, if a derivative is not in a polynomial equation, then we can find the order but degree cannot be determined.

The given differential equations is
${\left(\frac{dy}{dx}\right)}^3-4{\left(\frac{dy}{dx}\right)}^2+7y=sinx\Rightarrow {\left(\frac{dy}{dx}\right)}^3-4{\left(\frac{dy}{dx}\right)}^2+7y-sinx=0$
As the highest order derivative that occurs in the given differential equation is $\frac{dy}{dx},$ thus order of the equations is 1 and its degree is 3.
$(\because Thehighestpowerof\frac{dy}{dx},whichoccurs\left(\frac{dy}{dx}\right))$

Note:In any differential equation, if a derivative is not in a polynomial equation, then we can find the order but degree cannot be determined.

The given differential equations is
$\frac{d^{4}y}{d{x}^4}-sin\left(\frac{d^{3}y}{d{x}^3}\right)=0$
Since, the higest order derivative which occurs in the given differential equation is $\frac{d^{4}y}{d{x}^4}$. Thus, order of the given equation is 4, as $sin\left(\frac{d^{3}y}{d{x}^3}\right)$ occurs in the equation is not a polynomial equation. Hence, degree is not defined.

Note:In any differential equation, if a derivative is not in a polynomial equation, then we can find the order but degree cannot be determined.