1. For each of the differential equations given below, indicate its order and degree (if defined). (i) 2 2 2 5 6 log d y dy x y x dx dx (ii) 3 2 4 7 sin dy dy y x dx dx (iii) 4 3 4 3 sin 0 dy dy dx dx 2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation. (i) y = a ex + b ex + x 2 : 2 2 2 2 2 0 d y dy x xy x dx dx (ii) y = e x (a cos x + b sin x) : 2 2 2 20 d y dy y dx dx (iii) y = x sin 3x : 2 2 9 6cos3 0 d y y x dx (iv) x 2 = 2y 2 log y : 2 2 () 0 dy x y xy dx 3. Form the differential equation representing the famil
Question 1:For each of the differential equations given below, indicate its order and degree (if defined). (i) 2 2 2 5 6 log d y dy x y x dx dx ✩ ✪ ✭ ✮ ✫ ✬ ✯ ✰ (ii) 3 2 4 7 sin dy dy y x dx dx ✩ ✩ ✭ ✮ ✫ ✭ ✮ ✪ ✬ ✯ ✰ ✯ ✰ (iii) 4 3 4 3 sin 0 dy dy dx dx ✓ ✔ ✱ ✖ ✗ ✕
Question 2:For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation. (i) y = a ex + b e–x + x 2 : 2 2 2 2 2 0 d y dy x xy x dx dx (ii) y = e x (a cos x + b sin x) : 2 2 2 20 d y dy y dx dx (iii) y = x sin 3x : 2 2 9 6cos3 0 d y y x dx (iv) x 2 = 2y 2 log y : 2 2 () 0 dy x y xy dx
Question 3:Form the differential equation representing the family of curves given by (x – a) 2 + 2y 2 = a 2 , where a is an arbitrary constant.
Question 4:Prove that x 2 – y 2 = c (x 2 + y 2 ) 2 is the general solution of differential equation (x 3 – 3x y2 ) dx = (y 3 – 3x 2y) dy, where c is a parameter.
Question 5:Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Question 6:Find the general solution of the differential equation 2 2 1 0 1 dy y dx x .
Question 7:Show that the general solution of the differential equation 2 2 1 0 1 dy y y dx x x is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.
Question 8:Find the equation of the curve passing through the point 0, 4 whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Question 9:Find the particular solution of the differential equation (1 + e 2x ) dy + (1 + y 2 ) e x dx = 0, given that y = 1 when x = 0.
Question 10:Solve the differential equation 2 ( 0) x x y y y e dx x e y dy y ✓ ✔ ✕ ✖ ✗ ✚ ✘ ✛ .
Question 11:Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x
Question 12:Solve the differential equation 2 1( 0) x e y dx x x x dy✁ .
Question 13:Find a particular solution of the differential equation cot dy y x dx = 4x cosec x (x 0), given that y = 0 when 2 x .
Question 14:Find a particular solution of the differential equation (x + 1) dy dx = 2 e –y – 1, given that y = 0 when x = 0.
Question 15:The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
Question 16:The general solution of the differential equation 0 y dx x dy y is (A) xy = C (B) x = Cy 2 (C) y = Cx (D) y = Cx 2
Question 17:The general solution of a differential equation of the type P Q 1 1 dx x dy is (A) ✓ P P 1 1 Q C 1 dy dy y e e dy ✔ ✔ ✕ ✖ ✗ (B) ✘ P P 1 1 1 .Q C dx dx y e e dx ✚ ✚ ✛ ✜ ✢ (C) ✣ ✤ P P 1 1 Q C 1 dy dy x e e dy ✚ ✚ ✛ ✜ ✢ (D) ✥ P P 1 1 Q C 1 dx dx x e e dx ✔ ✔ ✕ ✖ ✗
Question 18:The general solution of the differential equation e x dy + (y ex + 2x) dx = 0 is (A) x ey + x 2 = C (B) x ey + y 2 = C (C) y ex + x 2 = C (D) y ey + x 2 =
