# IIT JAM Mathematics Test Series 2023 : Differential Equations – Linear Differential Equation

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## IIT JAM Mathematics Test Series 2023 : Differential Equations – Linear Differential Equation

Q1. Solve the Linear Differential Equation :

x cos x (dy/dx) + y (x sin x + cos x) = 1

• (a) yx sec x = tan x + c
• (b) xy sec y = tan x + c
• (c) sin x = tan x + c
• (d) yx sin x = tan x + c

Q2. Solve the Linear Differential Equation :

(1-x2) (dy/dx) + 2xy = x √(1-x2)

• (a) y = (1 – x)1/2
• (b) y = (1 – x2)1/2
• (c) y = (1 – x2)
• (d) y = (1 – x)

Q3. Solve the Linear Differential Equation :

sin x (dy/dx) + 3y = cos x

• (a) (y + 1/3) tan (x) = 2 tan (x) – x + c.
• (b) (y – 1/3) tan3 (x/2) = 2 tan3 (x/2) + x + c.
• (c) (y + 1/3) tan3 (x/2) = 2 tan (x/2) – x + c.
• (d) (y + 1/3) tan (x/2) = 2 tan (x/2) + x + c.

Q4. Integrate the differential Equation

(1+x2) (dy/dx) + 2xy – 4x2 = 0.

Also Obtain equation of the curve satisfying this equation and passing through the origin.

• (a) (1-x2) = (4/3) x3 + c and 4x3 = 3y (1+x2).
• (b) y (1-x2) = (4/3) x + c and 4x = 8y (1-x2).
• (c) (1+x2) = (4/3) x + c and 4x3 = 8y (1-x2).
• (d) y (1+x2) = (4/3) x3 + c and 4x3 = 3y (1+x2).

Q5. Solve the Linear Differential Equation :

(1 + y2) dx = (tan–1 y – x) dy

• (a) x = tan–1 y – 1 + cetan–1 y
• (b) x = cot–1 y + 1 + cecot–1 y
• (c) x = tan–1 y + 1 + cetan–1 y
• (d) x = cot–1 y – 1 + cecot–1 y

Q6. Solve the Linear Differential Equation :

(x + 1) (dy/dx) – ny = ex (x + 1)n+1

• (a) xy = tan-1y + c
• (b) xy = tany + c
• (c) xy = cot-1y + c
• (d) xy = coty + c

Q7. Solve the Linear Differential Equation :

dy/dx + y cos x = (1/2) × sin 2x

• (a) y = ce– sin x + sin x – 1
• (b) y = ce– cos x + sin x + 1
• (c) y = ce sin x + sin x – 1
• (d) y = ce cos x + sin x + 1

Q8. Solve the Linear Differential Equation :

(x log x) (dy/dx) + y = 2 log x

• (a) y log y = c – (log x)2
• (b) x log x = c + (log x)
• (c) x log y = c – (log x)
• (d) y log x = c + (log x)2

Q9. Solve the Linear Differential Equation :

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

• (a) y (sec x + tan x) = sec x + tan x – x + c
• (b) y (sec x – tan x) = sec x – tan x + x + c
• (c) y (sec x + tan x) = sec x + tan x + x + c
• (d) y (sec x – tan x) = sec x – tan x – x + c

Q10. Solve the Linear Differential Equation :

sin 2x (dy/dx) = y + tan x

• (a) y = sin x – c √(sinx)
• (b) y = tan x + c √(tanx)
• (c) y = sin x + c √(sinx)
• (d) y = tan x – c √(tanx)

Q11. Solve the Linear Differential Equation :

dy/dx + 2xy/(1+x2) = 1/(1+x2)

if y = 0, when x = 1

• (a) y (1 + x2) = tan–1 x – (π/4)
• (b) y (1 + x2) = tan–1 x – (π/4)
• (c) y (1 + x2) = tan–1 x – (π/4)
• (d) y (1 + x2) = tan–1 x – (π/4)

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