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

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


Q1. Solve the Homogeneous Differential Equations :

(x3 + 3xy2) dx + (y3 + 3x2y) dy = 0.

(a) (x2 + y2)2 + 4x2y2 = c
(b) (x2 – y2) – 4x2y2 = c
(c) (x2 + y2)2 – 8x2y2 = c
(d) (x2 – y2)2 – 4xy2 = c


Q2. Solve the Homogeneous Differential Equations :

x dy – y dx = (x2 + y2)1/2 dx

(a) xc = y + √(x2 – y2)
(b) x2c = y + √(x2 + y2)
(c) x2c = y – √(x2 – y2)
(d) x2c = y + √(x + y)


Q3. Solve the Homogeneous Differential Equations :

dy/dx = y/x + tan y/x

(a) cx = cos (y/x)
(b) cx = sin (y/x)
(c) cx = sin (x/y)
(d) cx = cos (x/y)


Q4. Solve the Homogeneous Differential Equations :

x cos (y/x) (y dx + x dy) = y sin (y/x) (x dy – y dx)

(a) y/x cos (y/x) = c
(b) x/y cos (y/x) = c
(c) xy cos (x/y) = c
(d) xy cos (y/x) = c


Q5. Solve the Homogeneous Differential Equations :

(4y + 3x) dy + (y – 2x) dx = 0

(a) c (x2 – 2xy – 2y2) = [{(√3+1)x + 2y}/{(√3-1)x – 2y} ]1/2√3
(b) c (x2 – 2xy + 2y2) = [{(√3+1)x + 2y}/{(√3-1)x + 2y} ]1/2√3
(c) c (x2 + 2xy + 2y) = [{(√3+1)x + 2y}/{(√3+1)x – 2y} ]1/2√3
(d) c (x2 + 2xy – y) = [{(√3+1)x + 2y}/{(√3+1)x – 2y} ]1/√3


Q6. Solve the Homogeneous Differential Equations :

x (dy/dx) = y {log y – log x + 1}

(a) y = xec
(b) y = xexc
(c) y = exc
(d) y = ec


Q7. Solve the Homogeneous Differential Equations :

{2√(xy) – x}dy + ydx = 0

(a) y = ce√(xy)
(b) y = ce(x/y)
(c) y = ce-√(x/y)
(d)y = ce(xy)


Q8. Solve the Homogeneous Differential Equations :

(x2 – 4xy – 2y2) dx + (y2 – 4xy – 2x2) dy = 0

(a) y3 – 6x2y + 6xy2 + x3 = c
(b) y3 – 6xy2 – 6x2y – x3 = c
(c) y3 + 6xy2 + 6x2y + x3 = c
(d) y3 – 6xy2 – 6x2y + x3 = c


Q9. Solve the Homogeneous Differential Equations :

x sin (y/x) (dy/dx) = y sin (y/x) – x

(a) x = cecos (y/x)
(b) x = cecos (yx)
(c) x = cesin (x/y)
(d) x = cesin (y/x)


Q10. Solve the Homogeneous Differential Equations :

y – x (dy/dx) = x + y (dy/dx)

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


Q11. Solve the Homogeneous Differential Equations :

(x3 – y3) dx + xy2 dy = 0

(a) 3x3/y3 logex = 1 + logec
(b) -3x3y3 logex = 1 – logec
(c) -3x3/y3 logex = 1 + logec
(d) 3x3y3 logex = 1 – logec


Q12. Solve the Homogeneous Differential Equations :

(x2 + xy) dy = (x2 + y2) dx

(a) (x + y)2 = c x eyx
(b) (x – y)2 = c x e–yx
(c) (x + y)2 = c x ey/x
(d) (x – y)2 = c x e–y/x


Q13. Solve the Homogeneous Differential Equations :

2 (dy/dx) = [y(x + y)/x2]

(a) (y – x)2 = cxy2
(b) (y + x) = cxy
(c) (y – x) = cxy
(d) (y + x)2 = cxy2


Q14. Solve the Homogeneous Differential Equations :

x2 dy + y(x + y) dx = 0

(a) y/x – 2 = cxy
(b) y/x + 2 = cxy
(c) y/x = cxy
(d) y/x + 2 = cx2y


Q15. Solve the Homogeneous Differential Equations :

(x2 + y2) dx – 2xdy = 0

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


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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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