# NCERT Solutions for Class 12 Maths Chapter 9 - Differential Equations - Exercise 9.1

### Access Exercises of Class 12 Maths Chapter 9 – Differential Equations

Exercise 9.1 Solutions 12 Questions

Exercise 9.2 Solutions 12 Questions

Exercise 9.3 Solutions 12 Questions

Exercise 9.4 Solutions 23 Questions

Exercise 9.5 Solutions 17 Questions

Exercise 9.6 Solutions 19 Questions

Miscellaneous Exercise on Chapter 9 Solutions 18 Questions

**Exercise 9.1**

**Determine order and degree (if defined) of differential equation given in Q. 1 to 18.**

$$\textbf{1.\space}\frac{\textbf{d}^{\textbf{4}}\textbf{y}}{\textbf{dx}^{\textbf{4}}} \textbf{+ sin}\textbf{(y"") = 0}\\\textbf{Sol.\space}\frac{d^{4}y}{dx^{4}} + \text{sin (y"")}=0\\\Rarr\space y"" + \text{sin(y"")}=0$$

The highest order derivative which occurs in the given differential equation is y'''', therefore, its order is 4.

As the given differential equation is not a polynomial equation in

$$\frac{dy}{dx}\space(i.e., y')\space\text{therefore, its degree}\\\text{is not defined.}$$

**2. y' + 5y = 0.**

**Sol.** The highest order derivative which occurs in the given differential equation is y' and index of highest power is one, therefore, the given differential equation is of order 1 and degree 1.

$$\textbf{3.\space}\bigg(\frac{\textbf{ds}}{\textbf{dt}}\bigg)^{\textbf{4}}\textbf{+ 3s}\frac{\textbf{d}^{\textbf{2}}\textbf{s}}{\textbf{dt}^{\textbf{2}}}\textbf{= 0.}$$

**Sol.** The highest order derivative present in the given differential equation is d^{2}s/dt^{2}. Therefore, the index of its highest power is one, hence the given differential equation is of order 2 and degree 1.

$$\textbf{4.\space}\bigg(\frac{\textbf{d}^{\textbf{2}}\textbf{y}}{\textbf{dx}^{\textbf{2}}}\bigg)^{\textbf{2}} \textbf{+ cos}\bigg(\frac{\textbf{dy}}{\textbf{dx}}\bigg)\textbf{= 0.}$$

**Sol.** The highest order derivative present in the given differential equation is

$$\frac{d^{2}y}{dx^{2}},\space\text{therefore, its order is 2.}$$

But the given differential equation is not a polynomial equation in dy/dx. Hence, its degree is not defined.

$$\textbf{5.\space}\frac{\textbf{d}^{\textbf{2}}\textbf{y}}{\textbf{dx}^{\textbf{2}}}\textbf{= cos 3x + sin 3x.} $$

**Sol.** The highest order derivative which occurs in the given differential equation is

$$\frac{d^{2}y}{dx^{2}}.\space\text{Therefore, its order is two.}\\\text{It is a polynomial equation in}\frac{d^{2}y}{dx^{2}}\\\text{and the power raised to}\frac{d^{2}y}{dx^{2}}\space\text{is 1.}$$

Hence, its degree is one.

**6. (y''') ^{2} + (y'')^{3} + (y')^{4} + y^{5} = 0.**

**Sol.** The highest order derivative which occurs in the given differential equation is y'''. Therefore, its order is three. The given differential equation is a polynomial equation in y''', y'' and y'.

The highest power raised to y''' is 2. Hence, its degree is 2.

**7. y''' + 2y'' + y' = 0.**

**Sol.** The highest order derivative which occurs in the given differential equation is y'''. Therefore, its order is three. It is a polynomial equation in y''', y'' and y'. The highest power raised to y''' is 1. Hence, its degree is 1.

**8. y' + y = e ^{x}.**

**Sol.** The highest order derivative present in the given differential equation is y'. Therefore, its order is one. The given differential equation is a polynomial equation in y'. The highest power raised to y' is 1. Hence, its degree is 1.

**9. y'' + (y') ^{2} + 2y = 0.**

**Sol.** The highest order derivative, present in the given differential equation is y''. Therefore, its order is two. The given differential equation is a polynomial equation in y'' and y' and the highest power raised to y'' is 1. Hence its degree is 1.

**10. y'' + 2y' + sin y = 0.**

**Sol.** The highest order derivative, present in the differential equation is y''. Therefore, its order is two.

This is a polynomial equation in y'' and y' and the highest power raised to y'' is one. Hence, its degree is one.

**11. The degree of the differential equation**

$$\bigg(\frac{\textbf{d}^{\textbf{2}}\textbf{y}}{\textbf{dx}^{\textbf{2}}}\bigg) \textbf{+} \bigg(\frac{\textbf{dy}}{\textbf{dx}}\bigg)^{2} \textbf{+ sin}\bigg(\frac{\textbf{dy}}{\textbf{dx}}\bigg)\textbf{+1}\textbf{= 0}\space\textbf{is :}$$

**(a) 3 (b) 2 (c) 1 (d) None of these**

**Sol.** (d) None of these

The given differential equation is not a polynomial equation in

$$\frac{dy}{dx}.\space\text{Therefore, its degree}\\\text{is not defined.}$$

**12. The order of the differential equation**

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

**(a) 2**

**(b) 1 **

**(c) zero **

**(d) None of these**

**Sol.** (a) 2

The highest order derivative present in the given differential equation is

$$\frac{d^{2}y}{dx^{2}}.\space\text{Therefore, its order is two.}$$