# NCERT Class 12 Maths Differential Equations

## Chapter 9 Ex.9.1 Question 1

Determine order and degree (if defined) of differential equation $$\frac{{{d^4}y}}{{d{x^4}}} + \sin \left( {y'''} \right) = 0$$.

### Solution

\begin{align} &\Rightarrow\; \frac{{{d^4}y}}{{d{x^4}}} + \sin \left( {y'''} \right) = 0\\ &\Rightarrow\; y'''' + \sin \left( {y'''} \right) = 0\end{align}

Highest order derivative in the differential equation is $$y''''$$. Its order is four.

Differential equation is not a polynomial equation in its derivatives. Its degree is not defined.

## Chapter 9 Ex.9.1 Question 2

Determine order and degree (if defined) of differential equation $$y' + 5y = 0$$.

### Solution

$$y' + 5y = 0$$

Highest order derivative in the differential equation is $$y'$$. Its order in one.

It is a polynomial equation in $$y'$$. Highest power $$y'$$ is $$1$$. Its degree is one.

## Chapter 9 Ex.9.1 Question 3

Determine order and degree (if defined) of differential equation $${\left( {\frac{{ds}}{{dt}}} \right)^4} + 3s\frac{{{d^2}s}}{{d{t^2}}} = 0$$.

### Solution

$${\left( {\frac{{ds}}{{dt}}} \right)^4} + 3s\frac{{{d^2}s}}{{d{t^2}}} = 0$$

Highest order derivative in the given differential equation is $$\frac{{{d^2}s}}{{d{t^2}}}$$. Its order is two.

It is a polynomial equation in $$\frac{{{d^2}s}}{{d{t^2}}}$$ and $$\frac{{ds}}{{dt}}$$.

The power $$\frac{{{d^2}s}}{{d{t^2}}}$$ is $$1$$. Degree is one.

## Chapter 9 Ex.9.1 Question 4

Determine order and degree (if defined) of differential equation $${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} + \cos \left( {\frac{{dy}}{{dx}}} \right) = 0$$.

### Solution

$${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} + \cos \left( {\frac{{dy}}{{dx}}} \right) = 0$$

Highest order derivative in the given differential equation is $$\frac{{{d^2}y}}{{d{x^2}}}$$. Order is $$2.$$

Given differential equation is not a polynomial equation in its derivatives. Degree is not defined.

## Chapter 9 Ex.9.1 Question 5

Determine order and degree (if defined) of differential equation $${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} = \cos 3x + \sin 3x$$.

### Solution

\begin{align}&{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} = \cos 3x + \sin 3x\\ &\Rightarrow\; {\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos 3x - \sin 3x = 0\end{align}

Highest order derivative in the given differential equation is $$\frac{{{d^2}y}}{{d{x^2}}}$$. Its order is two.

It is a polynomial equation in $$\frac{{{d^2}y}}{{d{x^2}}}$$ and the power is $$1$$. Its degree is $$1$$.

## Chapter 9 Ex.9.1 Question 6

Determine order and degree (if defined) of differential equation $${\left( {y'''} \right)^2} + {\left( {y''} \right)^3} + {\left( {y'} \right)^4} + {y^5} = 0$$.

### Solution

$${\left( {y'''} \right)^2} + {\left( {y''} \right)^3} + {\left( {y'} \right)^4} + {y^5} = 0$$

Highest order derivative present in the differential equation is $$y'''$$. Its order is three.

Given differential equation is a polynomial equation in $$y''',y''$$ and $$y'$$.

Highest power raised to $$y'''$$ is $$2$$. Its degree is $${\text{2}}$$.

## Chapter 9 Ex.9.1 Question 7

Determine order and degree (if defined) of differential equation $$y''' + 2y'' + y' = 0$$.

### Solution

$$y''' + 2y'' + y' = 0$$

Highest order derivative present in the differential equation is $$y'''$$. Its order is $$3.$$

It is a polynomial equation in $$y''',y''$$ and $$y'$$. The highest power $$y'''$$ is $$1$$. Its degree is $$1$$.

## Chapter 9 Ex.9.1 Question 8

Determine order and degree (if defined) of differential equation $$y' + y = e'$$.

### Solution

\begin{align}&y' + y = e'\\& \Rightarrow\; y' + y - e' = 0\end{align}

Highest order derivative present in the differential equation is $$y'$$. Its order is one.

Given differential equation is a polynomial equation in $$y'$$ and the highest power is one.

Its degree is one.

## Chapter 9 Ex.9.1 Question 9

Determine order and degree (if defined) of differential equation $$y'' + {\left( {y'} \right)^2} + 2y = 0$$.

### Solution

$$y'' + {\left( {y'} \right)^2} + 2y = 0$$

Highest order derivative present in the differential equation is $$y''$$. Its order is two.

Given differential equation is a polynomial equation in $$y''$$ and $$y'$$, the highest power$$y''$$ is one. Its degree is one.

## Chapter 9 Ex.9.1 Question 10

Determine order and degree (if defined) of differential equation $$y'' + 2y' + \sin y = 0$$.

### Solution

$$y'' + 2y' + \sin y = 0$$

Highest order derivative present in the differential equation is $$y''$$. Its order is two.

This is a polynomial equation in $$y''$$and $$y'$$ the highest power $$y''$$is one. Its degree is one.

## Chapter 9 Ex.9.1 Question 11

The degree of the differential equation $${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} + {\left( {\frac{{dy}}{{dx}}} \right)^2} + \sin \left( {\frac{{dy}}{{dx}}} \right) + 1 = 0$$ is

(A) $$3$$

(B) $$2$$

(C) $$1$$

(D) not defined

### Solution

$${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} + {\left( {\frac{{dy}}{{dx}}} \right)^2} + \sin \left( {\frac{{dy}}{{dx}}} \right) + 1 = 0$$

Differential equation is not a polynomial equation in its derivatives. Its degree is not defined.

Thus, the Correct option is D.

## Chapter 9 Ex.9.1 Question 12

The order of the differential equation $$2{x^2}\frac{{{d^2}y}}{{d{x^2}}} - 3\frac{{dy}}{{dx}} + y = 0$$.

(A) $$2$$

(B) $$1$$

(C) $$0$$

(D) not defined

### Solution

$$2{x^2}\frac{{{d^2}y}}{{d{x^2}}} - 3\frac{{dy}}{{dx}} + y = 0$$

Highest order derivative present in the given differential equation is $$\frac{{{d^2}y}}{{d{x^2}}}$$. Its order is two.

Thus, the correct option is A.

Differential Equations | NCERT Solutions
Instant doubt clearing with Cuemath Advanced Math Program