IIT JAM Mathematics Test Series 2023 : Differential Equations – Exact Differential Equations

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IIT JAM Mathematics Test Series 2023 : Differential Equations – Exact Differential Equations


Q1. Solve the Exact Differential Equation :

dx/dy + (ax+hy+g)/(hx+by+f) = 0

(a) ax2 + 2hxy + by2 + 2gx + 2fy – 2c = 0
(b) ax2 – 2hxy + by2 + 2gx – 2fy – 2c = 0
(c) ax2 + hxy + by2 + gx + fy – 2c = 0
(d) ax2 + 2hxy + by2 + 2fx + 2gy – 2c = 0


Q2. Solve the Exact Differential Equation :

(1 + ex/y ) dx + ex/y {1 – (x/y)} dy = 0

(a) x – yey/x = c
(b) x + yex/y = c
(c) x + yey/x = c
(d) x – yex/y = c


Q3. Solve the Exact Differential Equation :

(2x – y + 1) dx + (2y – x – 1) dy = 0

(a) x2 – xy – x – y2 – y = c
(b) x2 + xy + x + y2 + y = c
(c) x2 – xy + x – y2 – y = c
(d) x2 – xy + x – y2 = c


Q4. Solve the Exact Differential Equation :

(4x + 3y + 1) dx + (3x + 2y + 1) dy = 0

(a) x2 + xy + y2 – x – y = c
(b) x2 + xy + y2 + x + y = c
(c) 2x2 + 3xy + y2 – x – y = c
(d) 2x2 + 3xy + y2 + x + y = c


Q5. Solve the Exact Differential Equation :

dy/dx = (2x – y + 1)/(x + 2y – 3)

(a) x2 – xy + x + 3y – y2 = c
(b) x2 + xy + x + 3y + y2 = c
(c) x2 – xy + x – 3y – y = c
(d) x2 – xy + 3y – y2 = c


Q6. Solve the Exact Differential Equation :

(x + y)2 dx – (y2 – 2xy – x2) dy = 0

(a) x3 – y3 – 3xy (x – y) = c
(b) x3 + y3 + 3xy (x + y) = c
(c) x + y + 3xy (x + y) = c
(d) x3 – y3 + 3xy (x – y) = c


Q7. Solve the Exact Differential Equation :

{y (1 + 1/x) + cos y} dx + (x + log x – x sin y) dy = 0

(a) yx – y logx + x cosy = c
(b) xy + x logx + x cosy = c
(c) yx + y logx + x cosy = c
(d) yx + y logy + x cosx = c


Q8. Solve the Exact Differential Equation :

x dx + y dy + (x dx – y dy)/(x2+y2) = 0

(a) x2 + y2 + 2 tan–1 (x/y) = c
(b) x2 – y2 – 2 tan(x/y) = c
(c) x + y2 – 2 tan (x/y) = c
(d) x2 + y2 – 2 tan–1 (x/y) = c


Q9. Solve the Exact Differential Equation :

(r + sinθ – cosθ) dr + r (sinθ + cosθ) dθ = 0

(a) r2/2 + r (sinθ – θ) = c
(b) r/2 + r (cosθ – θ) = c
(c) r2/2 + r (sinθ + θ) = c
(d) r/2 – r (cosθ – θ) = c


Q10. Solve the Exact Differential Equation :

y sin 2x dx – (1 + y2 + cos2x) dy = 0

(a) 3y cos 2x + 9y – 2y = c
(b) 3y cos 2x + 9y + 2y3 = c
(c) 3y sin 2x + 9y + 2y3 = c
(d) 3y sin 2x + 9y + y3 = c


Q11. Solve the Exact Differential Equation :

(x2 + y2 + x) dx – (2x2 + 2y2 – y) dy = 0

(a) x + log (x2 – y2) – 2y = c
(b) x + 1/2 log (x2 + y2) = c
(c) x + 1/2 log (x2 + y2) – 2y = c
(d) x – log (x2 + y2) – 2y = c


Q12. Find the values of constant λ such that

(2xey + 3y2) (dy/dx) + (3x2 + λey) = 0

is exact. Further, for this value of λ, solve the equation.

(a) λ = 1 , x3 – 2ex – y3 = c
(b) λ = 3 , x3 – 2ex + y3 = c
(c) λ = 4 , x3 + 2ex – y3 = c
(d) λ = 2 , x3 + 2ex + y3 = c


Q13. Solve the Exact Differential Equation :

(a2 – 2xy – y2) dx – (x + y)2 dy = 0

(a) a2x – x2y – xy2 – (1/3) y3 = c
(b) a2x – x2y – xy2 – y3 = c
(c) ax – xy – xy2 – (1/3) y3 = c
(d) a2 – x2y – xy2 – y3 = c


Q14. Solve the Exact Differential Equation :

(ey + 1) cos x dx + eysin x dy = 0

(a) (ex + 1) sin y = c
(b) (ey + 1) sin x = c
(c) (ey – 1) sin x = c
(d) (ey – 1) cos x = c


Q15. Solve the Exact Differential Equation :

(x4 – 2xy2 + y4) dx – (2x2y – 4xy2 + sin y) dy = 0

(a) x/5 + x2y2 + xy4 + cos y = c
(b) x5/5 + xy – xy4 – sin y = c
(c) x5/5 – x2y2 + xy4 + cos y = c
(d) x5/5 – x2y2 – xy4 – sin y = c


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Answer Key

01.(a)06.(b)11.(c)
02.(b)07.(c)12.(d)
03.(c)08.(d)13.(a)
04.(d)09.(a)14.(b)
05.(a)10.(b)15.(c)

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