Table: Graph: The frequency of the function sin(x) is increased by 3 times.
Draw the graph of each of the following functions: Sin 3x
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Graphs of Trigonometric Functions
Draw the graph of each of the following functions: 3sin x
Table: Graph: The amplitude of the function sin(x) is increased by 3 times.
Draw the graph of each of the following functions: 2sin 3x
Table: Graph:
Draw the graph of each of the following functions: 2cos 3x
Table: Graph: The amplitude and frequency of the function cos(x) is increased by 2 and 3 times.
Draw the graph of each of the following functions: 
Table: Graph: The frequency of the function sin(x) is decreased by 0.5 times.
Draw the graphs of y = sin x and ![\begin{array}{l} y = \cos xin[0,2\pi ] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-523111e3d29d4af4c4181c8fa4667dd9_l3.png "Rendered by QuickLaTeX.com")
on the same axes.
Table: (i) For sinx (ii) For cosx Graph: The green line represents the curve for sin(x) and blue for cos(x) for...
Draw the graphs of y = cos x and ![\begin{array}{l} y = \cos 2xin[0,2\pi ] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-40846a1c50086ba5eac3bfb33637ca3a_l3.png "Rendered by QuickLaTeX.com")
on the same axes.
Table: (i) For cosx (ii) For cos(2x) Graph: The blue line depicts curve cos(2x) and the purple lines depict...
For each of the following differential equations, find a particular solution satisfying the given condition : 
, it being given that 
when 
Solution: On rearranging the terms we obtain: $\begin{array}{l} \frac{d y}{y}=\tan x d x \\ \Rightarrow \int \frac{d y}{y}=\int \tan x d x+c \\ \Rightarrow \log |y|=\log |\sec x|+\log c \end{array}$...
For each of the following differential equations, find a particular solution satisfying the given condition : 
, given thaty 
when 
Solution: On rearranging the terms we obtain: $\begin{array}{l} d y=\frac{2 x^{2}+1}{x} d x \\ \Rightarrow d y=2 x d x+\frac{1}{x} d x \end{array}$ On integrating both sides we obtain:...
For each of the following differential equations, find a particular solution satisfying the given condition: 
, it being given that 
when 
Solution: On rearranging the terms we obtain: $\frac{d y}{y^{2}}=-4 x d x$ On integrating both sides we obtain: $\Rightarrow \int \frac{d y}{y^{2}}=-\int 4 x d x+c$ $\Rightarrow...
For each of the following differential equations, find a particular solution satisfying the given condition: 
where 
and 
when
Solution: $\begin{array}{l} \cos \left(\frac{d y}{d x}\right)=a \\ \Rightarrow \frac{d y}{d x}=\cos ^{-1} a \\ \Rightarrow d y=\cos ^{-1} a d x \end{array}$ On integrating both sides we obtain:...
Find the general solution of each of the following differential equations: 
Solution: On rearranging the terms we obtain: $\frac{\cos x d x}{(1+\sin x)}=\frac{\sin y d y}{(1+\operatorname{cosy})}$ On integrating both the sides we obtain: $\Rightarrow \int \frac{\cos x d...
Find the general solution of each of the following differential equations: 
Solution: On rearranging the terms we obtain: $\frac{\sec ^{2} x d x}{\tan x}=-\frac{\sec ^{2} y d y}{\tan y}$ On integrating both sides we obtain: $\begin{array}{l} \Rightarrow \int \frac{\sec ^{2}...
Find the general solution of each of the following differential equations: 
Solution: On rearranging all the terms we obtain: $\frac{e^{x} d x}{1-e^{x}}=-\frac{\sec ^{2} y d y}{\operatorname{tany}}$ On integrating both sides we obtain: $\begin{array}{l} \Rightarrow \int...
Find the general solution of each of the following differential equations: 
Solution: $e^{2 x} e^{-3 y} d x+e^{2 y} e^{-3 x} d y=0$ On rearringing the terms we obtain: $\begin{array}{l} \Rightarrow \frac{e^{2 x} d x}{e^{-3 x}}=-\frac{e^{2 y} d y}{e^{-3 y}} \\ \Rightarrow...
Find the general solution of each of the following differential equations: 
Solution: $\begin{array}{l} \text { It is given that: } \frac{d y}{d x}=e^{x} e^{-y}+x^{2} e^{-y} \\ \Rightarrow \frac{d y}{d x}=e^{-y}\left(e^{x}+x^{2}\right) \\ \Rightarrow \frac{d...
Find the general solution of each of the following differential equations: 
Solution: $\begin{array}{l} \left(e^{x}+ e^{-x}\right) d y-\left(e^{x}-e^{-x}\right) d x=0 \\ \Rightarrow d y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x \end{array}$ On integrating both sides we obtain,...
Find the general solution of each of the following differential equations: 
Solution: $\frac{d y}{d x}=e^{x} e^{y}$ On rearringing the terms we obtain: $\Rightarrow \frac{d y}{e^{y}}=e^{x} d x$ On integrating both sides we obtain, $\begin{array}{l} \Rightarrow \int \frac{d...
Find the general solution of each of the following differential equations: 
Solution: $(x-1) \frac{d y}{d x}=2 x^{3} y$ On separating the variables we obtain: $\begin{array}{l} \Rightarrow \frac{d y}{y}=2 x^{3} \frac{d x}{(x-1)} \\ \Rightarrow \frac{d...
Find the general solution of each of the following differential equations: 
Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=1-x+y-x y=1+y-x(1+y) \\ \Rightarrow \frac{d y}{d x}=(1+y)(1-x) \end{array}$ On rearranging the terms we obtain: $\Rightarrow \frac{d...
Find the general solution of each of the following differential equations: 
Solution: $\begin{array}{l} \frac{d y}{d x}=1+x+y+x y=1+y+x(1+y) \\ \Rightarrow \frac{d y}{d x}=(1+y)(1+x) \end{array}$ On rearranging the terms we obtain: $\Rightarrow \frac{d y}{1+y}=(1+x) d x$ On...
Find the general solution of each of the following differential equations: 
Solution: $\begin{array}{l} x^{4} \frac{d y}{d x}=-y^{4} \\ \Rightarrow \frac{d y}{-y^{4}}=\frac{d x}{x^{4}} \end{array}$ On integrating both sides we obtain, $\begin{array}{l} \Rightarrow \int...
Find the general solution of each of the following differential equations: 
Solution: $\frac{d y}{d x}=\left(1+x^{2}\right)\left(1+y^{2}\right)$ On rearranging the terms,we obtain: $\Rightarrow \frac{d y}{1+y^{2}}=\left(1+x^{2}\right) d x$ On integrating both sides we...