In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve each of them. 1. (x 2 + xy) dy = (x 2 + y 2 ) dx 2. x y y x 3. (x y) dy (x + y) dx = 0 4. (x 2 y 2 ) dx + 2xy dy = 0 5. 2 22 2 dy x x yxy dx 6. x dy y dx = 2 2 x y dx 7. cos sin sin cos yy yy x y y dx y xx dy xx xx 8. sin 0 dy y x yx dx x 9. log 2 0 y y dx x dy x dy x 10. 1 1 0 x x y y x e dx e dy y For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition: 11. (x + y) dy + (x y) dx = 0; y = 1 when x = 1 12. x 2 dy + (xy + y 2 ) dx
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve each of them.
Question 1:(x 2 + xy) dy = (x 2 + y 2 ) dx
Question 2:x y y x✁
Question 3:(x – y) dy – (x + y) dx = 0
Question 4:(x 2 – y 2 ) dx + 2xy dy = 0
Question 5:2 22 2 dy x x yxy dx
Question 6:x dy – y dx = 2 2 x y dx
Question 7:cos sin sin cos yy yy x y y dx y xx dy xx xx ✓ ✓
Question 8:sin 0 dy y x yx dx x ✔ ✕ ✖ ✗ ✚ ✘ ✛ ✜
Question 9:log 2 0 y y dx x dy x dy x ✔ ✕ ✗ ✚ ✖ ✘ ✛ ✜
Question 10:1 1 0 x x y y x e dx e dy y ✢ ✣ ✢ ✣ ✤ ✤ ✥ ✩ ✪ ✩ ✪ For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
Question 11:(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
Question 12:x 2 dy + (xy + y 2 ) dx = 0; y = 1 when x = 1
Question 13:2 sin 0; 4 ✫ ✮ ✯ ✬ ✭ ✳ ✴ ✰ ✱ ✲ ✲ ✵ ✶ ✷ ✸ ✹ ✺ y x y dx x dy y x when x = 1
Question 14:cosec 0 dy y y dx x x ✔ ✕ ✖ ✗ ✚ ✘ ✛ ✜ ; y = 0 when x = 1
Question 15:2 2 2 20 dy xy y x dx ; y = 2 when x = 1
Question 16:A homogeneous differential equation of the from dx x h dy y ✻ ✼ ✽ ✾ ✿ ❀ ❁ can be solved by making the substitution. (A) y = vx (B) v = yx (C) x = vy (D) x
Question 17:Which of the following is a homogeneous differential equation? (A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0 (B) (xy) dx – (x 3 + y 3 ) dy = 0 (C) (x 3 + 2y 2 ) dx + 2xy dy = 0 (D) y 2 dx + (x 2 – xy – y 2 ) dy = 0
