Summary - Mathematics

.
Chapter 9 STD 12 Date : 16/12/25
Differential Equations Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [30]
é 2 æ yö ù p
1. find the particular solution satisfying the given condition : ê x sin çè x ÷ø - y ú dx + x dy = 0; y = when x = 1.
ë û 4
2. For the differential equation in exercise find the general solution : sec2x tan y dx + sec2y tan x dy = 0
dy
3. For the differential equation xy = (x + 2)(y + 2), find the solution curve passing through the point
dx
(1, –1).
dy
4. x + 2y = x2 log x (x > 0)
dx
æ pö
5. Find the equation of the curve passing through the point çè 0, 4 ÷ø whose differential equation is sin x
cos y dx + cos x sin y dy = 0.
6. Find the particular solution of the differential equation (1 + e2x)dy + (1 + y2)ex dx = 0, given that
y = 1 when x = 0.
dy
7. For the differential equation in exercise find the general solution : = (1 + x 2 )(1 + y 2 )
dx
8. y dx + (x – y2) dy = 0

dy 1 - y2
9. Find the general solution of the differential equation + = 0.
dx 1 - x2
dy
10. + 3y = e–2x
dx
dy
11. For the differential equation in exercise find the general solution : = 4 - y 2 (–2 < y < 2)
dx
12. For the differential equation, find a particular solution satisfying the given condition :

æ dy ö
cos çè dx ÷ø = a (a Î R); y = 1 when x = 0.
dy
13. For the differential equation, find a particular solution satisfying the given condition : = y tan
dx
x; y = 1 when x = 0.
dy 1 - cos x
14. For the differential equation in exercise find the general solution : dx = 1 + cos x
ée – 2 x y ù dx
15. Solve the differential equation ê x - x ú dy = 1 ( x ¹ 0).
ëê úû

//X Section B
• Write the answer of the following questions. [Each carries 4 Marks] [68]
16. In a bank, principal increases continuously at the rate of 5% per year. In how many years
` 1000 double itself ?
x x
17. Show that the differential equation 2y e y dx + (y – 2x e y ) dy = 0 is homogeneous and find
its particular solution, given that, x = 0 when y = 1.
18. In a bank, principal increases continuously at the rate of 5% per year. An amount of ` 1000 is
deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648). #
19. In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many
hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number
present ?

, In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many
hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number
present ?
dy
20. Find a particular solution of the differential equation + y cot x = 4x cosec x (x ¹ 0), given that
dx
p
y = 0 when x = .
2
x x
21. Solve the differential equation ye y dx = ( x e y + y 2 ) dy (y ¹ 0).
22. For the differential equation, find a particular solution satisfying the given condition :

dy
(x3 + x2 + x + 1) = 2x2 + x; y = 1 when x = 0.
dx
æ y ö dy æ yö
23. Show that the differential equation x cos çè ÷ø = y cos çè ÷ø + x is homogeneous and solve it.
x dx x
24. show that the given differential equation is homogeneous and solve it :


{ x cos æç ö÷ + y sin æç ö÷
y
è xø
y
è xø } { æ yö æ yö
}
y dx = y sin çè x ÷ø - x cos çè x ÷ø x dy

25. Prove that x2 – y2 = c(x2 + y2 )2 is the general solution of differential equation (x3 – 3xy2 )
dx = (y3 – 3x2y)dy, where c is a parameter.
26. Solve the differential equation :
(tan–1y – x)dy = (1 + y2)dx
dy
27. Find a particular solution of the differential equation (x + 1) = 2e–y –1, given that y = 0 when
dx
x = 0.
28. Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates
of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that
point by 5.
dy
29. Find the particular solution of the differential equation + y cot x = 2x + x2 cot x (x ¹ 0) given
dx
p
that y = 0 when x = .
2
30. Solve the differential equation :

æ yö æ yö
(x dy – y dx)y sin çè ÷ø = (y dx + x dy)x cos çè ÷ø .
x x
x x
æ1- xö
31. show that the given differential equation is homogeneous and solve it : (1 + e ) dx + e y y çè y ÷ø

dy = 0
æ yö
32. show that the given differential equation is homogeneous and solve it : y dx + x log çè x ÷ø dy – 2x dy = 0

, .
Chapter 9 STD 12 Date : 16/12/25
Differential Equations Maths

Section [ A ] : 2 Marks Questions

No Ans Chap Sec Que Universal_QueId
1. - Chap 9 [Part-2] S4 13 QP25P11B1213_P2C9S4Q13
2. - Chap 9 [Part-2] S3 4 QP25P11B1213_P2C9S3Q4
3. - Chap 9 [Part-2] S3 16 QP25P11B1213_P2C9S3Q16
4. - Chap 9 [Part-2] S5 6 QP25P11B1213_P2C9S5Q6
5. - Chap 9 [Part-2] S6 6 QP25P11B1213_P2C9S6Q6
6. - Chap 9 [Part-2] S6 7 QP25P11B1213_P2C9S6Q7
7. - Chap 9 [Part-2] S3 6 QP25P11B1213_P2C9S3Q6
8. - Chap 9 [Part-2] S5 11 QP25P11B1213_P2C9S5Q11
9. - Chap 9 [Part-2] S6 4 QP25P11B1213_P2C9S6Q4
10. - Chap 9 [Part-2] S5 2 QP25P11B1213_P2C9S5Q2
11. - Chap 9 [Part-2] S3 2 QP25P11B1213_P2C9S3Q2
12. - Chap 9 [Part-2] S3 13 QP25P11B1213_P2C9S3Q13
13. - Chap 9 [Part-2] S3 14 QP25P11B1213_P2C9S3Q14
14. - Chap 9 [Part-2] S3 1 QP25P11B1213_P2C9S3Q1
15. - Chap 9 [Part-2] S6 10 QP25P11B1213_P2C9S6Q10


Section [ B ] : 4 Marks Questions

No Ans Chap Sec Que Universal_QueId
16. - Chap 9 [Part-2] S7 9 QP25P11B1213_P2C9S7Q9
17. - Chap 9 [Part-2] S7 12 QP25P11B1213_P2C9S7Q12
18. - Chap 9 [Part-2] S3 21 QP25P11B1213_P2C9S3Q21
19. - Chap 9 [Part-2] S3 22 QP25P11B1213_P2C9S3Q22
20. - Chap 9 [Part-2] S6 11 QP25P11B1213_P2C9S6Q11
21. - Chap 9 [Part-2] S6 8 QP25P11B1213_P2C9S6Q8
22. - Chap 9 [Part-2] S3 11 QP25P11B1213_P2C9S3Q11
23. - Chap 9 [Part-2] S7 11 QP25P11B1213_P2C9S7Q11
24. - Chap 9 [Part-2] S4 7 QP25P11B1213_P2C9S4Q7
25. - Chap 9 [Part-2] S6 3 QP25P11B1213_P2C9S6Q3
26. - Chap 9 [Part-2] S7 22 QP25P11B1213_P2C9S7Q22
27. - Chap 9 [Part-2] S6 12 QP25P11B1213_P2C9S6Q12
28. - Chap 9 [Part-2] S5 17 QP25P11B1213_P2C9S5Q17
29. - Chap 9 [Part-2] S7 17 QP25P11B1213_P2C9S7Q17

Welcome To Future - Quantum Paper