For each of the differential equations in Exercises 1 to 10, find the general solution: 1. 1 cos 1 cos dy x dx x 2. 2 4 ( 2 2) dy y y d 3. 1 ( 1) dy y y dx 4. sec2 x tan y dx + sec2 y tan x dy = 0 5. (e x + e x ) dy (e x e x ) dx = 0 6. 2 2 (1 ) (1 ) dy x y dx 7. y log y dx x dy = 0 8. 5 5 dy x y dx 9. 1 sin dy x dx 10. e x tan y dx + (1 e x ) sec2 y dy = 0 For each of the differential equations in Exercises 11 to 14, find a particular solution satisfying the given condition: 11. 3 2 ( 1) dy xxx dx = 2x 2 + x; y = 1 when x = 0 12. 2 ( 1) 1 dy x x dx ; y = 0 when x = 2 13. cos dy a dx (a R); y = 1 when x = 0 14. tan dy y x
For each of the differential equations in Exercises 1 to 10, find the general solution:
Question 1:1 cos 1 cos dy x dx x
Question 2:2 4 ( 2 2) dy y y d
Question 3:1 ( 1) dy y y dx
Question 4:sec2 x tan y dx + sec2 y tan x dy = 0
Question 5:(e x + e –x ) dy – (e x – e –x ) dx = 0
Question 6:2 2 (1 ) (1 ) dy x y dx
Question 7:y log y dx – x dy = 0
Question 8:5 5 dy x y dx
Question 9:1 sin dy x dx
Question 10:e x tan y dx + (1 – e x ) sec2 y dy = 0 For each of the differential equations in Exercises 11 to 14, find a particular solution satisfying the given condition:
Question 11:3 2 ( 1) dy xxx dx = 2x 2 + x; y = 1 when x = 0
Question 12:2 ( 1) 1 dy x x dx ; y = 0 when x = 2
Question 13:cos dy a dx (a R); y = 1 when x = 0
Question 14:tan dy y x dx ; y = 1 when x = 0
Question 15:Find the equation of a curve passing through the point (0, 0) and whose differential equation is y = e x sin x.
Question 16:For the differential equation ( 2) ( 2) dy xy x y dx , find the solution curve passing through the point (1, –1).
Question 17:Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
Question 18:At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).
Question 19:The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds
Question 20:In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge 2 = 0.6931).
Question 21:In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e 0 5 = 1.648).
Question 22:In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
Question 23:The general solution of the differential equation dy x y e dx✁ is (A) e x + e –y = C (B) e x + e y = C (C) e –x + e y = C (D) e –x + e –y =
