Solve the following differential equations: dx+ ^2y dy=0 - Vidyadip

Question

Solve the following differential equations:

$\tan\text{y dx}+\sec^2\text{y}\tan\text{x dy}=0$

✓

Answer

We have,
$\tan\text{y dx}+\sec^2\text{y}\tan\text{x dy}=0$
$\Rightarrow\sec^2\text{y}\tan\text{x dy}=-\tan\text{y dx}$
$\Rightarrow\frac{\sec^2\text{y}}{\tan\text{y}}\text{dy}=-\frac{1}{\tan\text{x}}\text{dx}$
$\Rightarrow\frac{1}{\cos^2\text{y}}\times\frac{\cos\text{y}}{\sin\text{y}}\text{dy}=-\cot\text{x dx}$
$\Rightarrow\frac{1}{\sin\text{y}\cos\text{y}}\text{dy}=-\cot\text{x dx}$
$\Rightarrow\frac{2}{\sin2\text{y}}\text{dy}=-\cot\text{x dx}$
$\Rightarrow2\text{ cosec }2\text{y dy}=-\cot\text{x dx}$
Integrating both sides, we get
$2\int\text{cosec}\text{ 2y dy}=-\int\cot\text{x dx}$
$\Rightarrow\log\tan\text{x}=-\log\sin\text{x}=\log\text{C}$
$\Rightarrow\log \tan\text{x}+\log\sin\text{x}=\log\text{C}$
$\Rightarrow\log(\tan\text{ x}\times\sin\text{x})=\log\text{C}$
$\Rightarrow\tan\text{x}\times\sin\text{x}=\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If x^y + y^x = (x + y)^x+y, find $\frac{\text{dy}}{\text{dx}}$

Solve the following differential equations:

$\cos\text{x}\cos\text{y}\frac{\text{dy}}{\text{dx}}=-\sin\text{x}\sin\text{y}$

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

	A	B
Grinding	1	2
Turning	3	1
Assembling	6	3
Testing	5	4

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

$\int\frac{\text{x}^2+3\text{x}+1}{(\text{x}+1)^2}\text{dx}$

A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs. 20 per unit for the product A and Rs. 15 per unit for the product B, how many units of each product should be sold to maximize profit?

Let R be a relation on the set N given by R = {(a, b): a = b - 2, b > 6}. Then,

(2, 4) ∈ R
(3, 8) ∈ R
(6, 8) ∈ R
(8, 7) ∈ R

Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$

If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ with $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

A trust fund has ₹ 30, 000 that must be inverted in two different types of bond. the first bond pays 5% interest per year and the second bond pays 7% interest per year. using matrix multiplication, determine how to divide ₹ 30,000 in two types of bonds, if the trust fund must obtain an annual interest of (a) ₹ 1800, (b) ₹ 2000.

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.