Order and Degree of a Differential Equation MCQ Quiz in मल्याळम - Objective Question with Answer for Order and Degree of a Differential Equation - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept - 

Order of a Differential Equation: The order of a differential equation is the highest power of the derivative present in the equation.

Degree of a Differential Equation: The degree of a differential equation, when it is defined, is the exponent of the highest order derivative in the equation after it has been simplified as much as possible. 

Calculation:

Given the differential equation:

${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^{5/6}={\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^{1/3}$

taking power '6' on both sides, we get-

${\left({\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^{5/6}\right)}^6={\left[{\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^{1/3}\right]}^6$

⇒ ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^5={\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^2$

⇒ ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^5-{\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^2$ = 0

So, from above it quite clear that Order = 2, Degree = 5

The correct answer is option "2'

Concept:

Order: The order of a differential equation is the highest power of the derivative present in the equation.

Degree: The degree of a differential equation, when it is defined, is the exponent of the highest order derivative in the equation after it has been simplified as much as possible. 

Solution:

The differential equation  ${\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^4+3\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}$ = 0

The highest-order derivative is $\frac{d^{2}y}{d{x}^2}$ whose power is 1.

Therefore, the degree of the differential equation ${\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^4+3\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}$ = 0 is 1

Hence, option 4 is correct.

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given:

$\mathrm{y}{\left({\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}\right)}^3=\mathrm{x}{\displaystyle \frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}}$

For the given differential equation the highest order derivative is 2.

Now, the power of the highest order derivative is 1.

We know that, the degree of a differential equation is the power of the highest derivative

Hence, the degree of the differential equation is 1.

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given:

$$\begin{gathered} \mathrm{y}=\mathrm{x}{\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}+{\left({\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}\right)}^{-1} \\ \mathrm{y}=\mathrm{x}{\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}+\left({\displaystyle \frac{1}{{\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}}}\right) \\ \mathrm{y}{\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}=\mathrm{x}({\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}}{)}^2+1 \end{gathered}$$

For the given differential equation the highest order derivative is 1.

Now, the power of the highest order derivative is 2.

We know that, the degree of a differential equation is the power of the highest derivative

Hence, the degree of the differential equation is 2.

Concept:

The order of differential equation is the order of the highest derivative appearing in it.

The degree of a differential equation is the degree of the highest derivative occurring in it, after the equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

The given differential equation can be rewritten as

${\left[\frac{d^{2}y}{d{x}^2}\right]}^4=y+{\left(\frac{dy}{dx}\right)}^2$

Here, the power of highest order derivative $\left(i.e.,\frac{d^{2}y}{d{x}^2}\right)$ is 4.

Concept:

The order of a differential equation is the order of the highest derivative appearing in it.

The degree of a differential equation is the degree of the highest derivative occurring in it, after the equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given the differential equation is

4x${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^2$ + $\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}$ = $\sqrt{{\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^3+{\mathrm{y}}^8}$

Squaring both sides, we get

$[4\mathrm{x}(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}{)}^2+\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{]}^2=(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{)}^3+{\mathrm{y}}^8$

$16{\mathrm{x}}^2(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}{)}^4+8\mathrm{x}(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}{)}^2\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}+{\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)}^2=(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{)}^3+{\mathrm{y}}^8$

The power of $\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}$ is 4

Here, the highest derivative is $\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}$

So, the order of the given differential equation = 2

The power of the highest derivate = 4

So, the degree of the given differential equation = 4

Concept:

Order of a differential equation: The order of a differential equation is the order of the highest order derivative appearing in the equation.

Degree of a differential equation: The degree of a differential equation is the degree of the highest order derivative when differential coefficients are made free from radicals and fractions.

Calculation:

We have, $\frac{d^{3}x}{d{t}^3}+{\left(\frac{d^{2}x}{d{t}^2}\right)}^7+{\left(\frac{dx}{dt}\right)}^5=e^t$

Clearly, the highest order differential coefficient in this equation is $\frac{d^{3}x}{d{t}^3}$ and its power is 1. Therefore, the given differential equation has an order of 3 and a degree of 1.

Hence, the order is 3 and the degree is 1.

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given the differential equation is $\frac{{\mathrm{d}}^3\mathrm{y}}{\mathrm{d}{\mathrm{x}}^3}+\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}\right)-2=0$

The given differential equation is not a polynomial equation in its derivatives.

Hence its degree is not defined

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

$\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}-\mathrm{x}=(\mathrm{y}-\mathrm{x}\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{)}^{-6}$, rearranging the given equation we get,

$\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}-\mathrm{x}=\frac{1}{(\mathrm{y}-\mathrm{x}\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{)}^6}$

$(\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}-\mathrm{x})\times (\mathrm{y}-\mathrm{x}\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}{)}^6=1$

After multiplying the two terms the power of the highest derivative will be 7

Hence the degree = 7

Concept:

The order of a differential equation is the order of the highest derivative appearing in it.

The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

$\Rightarrow \frac{d^{2}y}{d{x}^2}+2{\left(\frac{dy}{dx}\right)}^2+9y=x$

⇒ In above differential equation highest derivative is two and its power is one

∴ Order = 2 and Degree  = 1

Both 1 and 2 are correct.