# IIT JAM Mathematics Test Series 2023 : Differential Equations – Integrating Factor

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## IIT JAM Mathematics Test Series 2023 : Differential Equations – Integrating Factor

Q1. Find Integrating Factor by of the given Differential Equation :

y (1 + xy) dx + x (1 – xy) dy = 0

(a) 1/(2x2y2).
(b) (2x2y2)
(c) (x2y2)
(d) 1/(x2y2)

Q2. Find Integrating Factor by of the given Differential Equation :

(xy sin xy + cos xy) y dx + (xy sin xy – cos xy) x dy = 0

(a) (2xy cos xy)
(b) 1/(2xy cos xy)
(c) (2xy sin xy)
(d) 1/(2xy sin xy)

Q3. Find Integrating Factor by of the given Differential Equation :

(x3y3 + x2y2 + xy + 1) y dx + (x3y3 – x2y2 – xy + 1) xdy = 0.

(a) {2x2y2 (xy + 1)}
(b) 1/{x2y2 (xy – 1)}
(c) 1/{2x2y2 (xy + 1)}
(d) {x2y2 (xy – 1)}

Q4. Solve the given Differential Equation :

y (1 – xy) dx – x (1 + xy) dy = 0

(a) log (x/y) + xy = c
(b) log (x/y) – x/y = c
(c) log (x/y) +x/y = c
(d) log (x/y) – xy = c

Q5. Find Integrating Factor by of the given Differential Equation :

(x2 + y2 + x) dx + xy dy = 0

(a) elnx
(b) lnx
(c) e-lnx
(d) -lnx

Q6. Solve the given Differential Equation :

(5xy + 4y2 + 1)dx + (x2 + 2xy)dy = 0

(a) x5y – x4y – x4/4 = 0
(b) x5y + x4y2 + x4/4 = 0
(c) x5y – x4y2 + x4/4 = 0
(d) x5y + x4y2 – x/4 = 0

Q7. Solve the given Differential Equation :

(x2 + y2 + 2x)dx + 2ydy = 0

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

Q8. Solve the given Differential Equation :

(xy2 + 2x2y3) dx + (x2y – x3y2) dy = 0.

(a) log (x2y) + (1/xy) = c
(b) log (xy) – (1/xy) = c
(c) log (x2/y) + (xy) = c
(d) log (x2/y) – (1/xy) = c

Q9. Find Integrating Factor by of the given Differential Equation :

(xy2 – x2)dx + (3x2y2 + x2y – 2x3 + y2)dy = 0

(a) e6y
(b) 6y
(c) e-6y
(d) -6y

Q10. Find Integrating Factor by of the given Differential Equation :

xα yβ (my dx + nx dy) = 0,

(a) xm–1+α yn–1–β
(b) xkm–1–α ykn–1–β
(c) xkm+1+α ykn+1+β
(d) xk–1–α yk–1+β

Q11. Find Integrating Factor by of the given Differential Equation :

(y2 + 2x2y)dx + (2x3– xy)dy = 0

(a) x2/5 y2/1
(b) x–2/5 y–2/1
(c) x–5/2 y–1/2
(d) x5/2 y1/2

Q12. Solve the given Differential Equation :

(x4y4 + x2y2 + xy) ydx + (x4y4 – x2y2 + xy) xdy = 0

(a) x2y2 + (1/xy) – log (x/y) = c
(b) (1/2) x2y2 – (1/xy) + log (x/y) =
(c) x2y2 + (1/xy) – log (x/y) =
(d) (1/2) x2y2 – (1/xy) + log (x/y) = c

Q13. Find Integrating Factor by of the given Differential Equation :

(2ydx + 3xdy) + 2xy (3ydx + 4xdy) = 0

(a) xy2
(b) xy
(c) x2y
(d) x2y2

Q14. Given that the differential equation

(2x2y2 + y) dx – (x3y – 3x) dy = 0

has an I.F. of the form xh yk , find its general solution.

(a) 2x10/7 y-5/7 + 10x4/7 y-12/7 = c
(b) 4x10/7 y-5/7 – 5x-4/7 y-12/7 = c
(c) 2x10/7 y-5/7 + 10x-4/7 y-12/7 = c
(d) 4x10/7 y-5/7 – 5x4/7 y-12/7 = c

Q15. Solve the given Differential Equation :

(x2y2 + xy + 1) y dx + (x2y2– xy + 1) xdy = 0

(a) xy + (xy) + log (x/y) = c
(b) xy – (xy) – log (x/y) = c
(c) xy – (1/xy) + log (x/y) = c
(d) xy + (1/xy) – log (x/y) = c

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