MP Board Class 12th Maths Important Questions Chapter 9 Differential Equations

MP Board Class 12th Maths Important Questions Chapter 9 Differential Equations

MP Board Class 12th Maths Important Questions Chapter 9 Differential Equations

Important Questions

Very Short Answer Type Questions

Question 1.
Find the order and degree of  + y = e^-x?
Answer:
1, 1

Question 2.
Find the degree and order of  ?
Answer:
1, 6

Question 3.
Find the degree and order of  ?
Answer:
2, 2

Question 4.
Find the differential equation corresponding to equation of circle x² + y² = a²?
Answer:
y  + x = 0

Question 5.
Find the differential equation of line y = mx + c?
Answer:
= m

Question 6.
Solve the differential equation  = 4y?
Answer:
y = c.e^4x

Question 7.
Solve the differential equation x²  = 2?
Answer:
y = c –

Question 8.
Find the solution of differential equation dy = sinxdx?
Answer:
y + cos x = c

Question 9.
Find the solution of differential equation  + Px = Q?
Answer:
xe∫^pdy. dy + c

Question 10.
Solve the differential equation (1 – y²)  + yx = ay?
Answer:

Short Answer Type Questions

Question 1.
Solve the differential equation xlog xdy – ydx = 0?
Solution:
x log x dy – ydx = 0 (given)
⇒ x log x = ydx

Question 2.
Prove that solution of y = 4sin 3x is   + 9y = 0?
Solution:
y = 4 sin 3x (given)
Differentiating with respect to x,
∴   = 12 cos 3x
Again differentiating with respect to x,
= – 36 sin 3x = – 9 × 4 sin 3x
⇒  = -9y, [from eqn.(1)]
⇒  + 9y = 0. Proved.

Question 3.
Solve the differential equation  = sec² x + 2x?
Solution:
= sec x (sec x + tan x) (given)
⇒ dy = (sec² x + sec x tan x ) dx
⇒ ∫dy = ∫sec² dx + ∫sec x tan x dx
∴y = tan x + sec x + c.

Question 4.
Solve the differential equation  = sec² x + 3x²?
Solution:
= sec² x + 3x² (given)
⇒ dy = (sec² x + 3x²) dx
⇒ ∫dy = ∫sec² xdx + 3∫x² dx
⇒ y = tan x +  + c
⇒ y = tan x + x³ + c.

Question 5.
Solve the differential equation  = sec² x + 2x?
Solution:
Solve as Q.No . 4

Question 6.
Solve the differential equation  = (3x² x + 2)?
Solution:
= (3x² + 2) (given)
⇒ dy = (3x² + 2) dx
⇒ ∫dy = ∫(3x² + 2) dx
⇒ y = 3 ×  + 2x = c = x² + 2x + c.

Question 7.
Solve the differential equation x²  = 2?
Solution:

Question 8.
Solve the differential equation  = x³ + sin 4x?
Solution:

Question 9.
Solve the differential equation dydx + 2x = e^3x?
Solution:

Question 10.
Solve the differential equation  = sin x sin y?
Solution:
= sin x sin y
⇒ cosec y dy = sin x dx
On integrating,
– log_e (cosec y + cot y) = – cos x + c
⇒ cos x – log_e (cosec y + cot y) = c

Question 11.
Solve the differential equation  = y sin x?
Solution:

Question 12.
Solve the differential equation  = x cos x?
Solution:
Given:
= x cos x
⇒ dy = x cos x dx
⇒ ∫dy = ∫x cos x dx
⇒ y = xsin x – ∫1. sin x dx + c
⇒ y = xsin x + cos x + c

Question 13.
Solve the differential equation  = 1 – x + y – xy?
Solution:

Question 14.
Solve the differential equation  = (1 + x) (1 + y²)?
Solution:

Question 15.
Solve the differential equation:
= cot² x?
Solution:
= cot² x (given)
⇒ dy = cot² x
⇒ ∫dy = ∫cot² x dx
⇒ y = ∫(cosec² x – 1) dx
⇒ y = – cot x – x + c

Long Answer Type Questions

Question 1.
(A) Solve the differential equation  + y tan x = sec x?
Solution:
+ y tan x = sec x (given)
Comparing the equation with  + Py = Q,
P = tan x, Q = sec x
∴L.F. = e^{∫P dx} = e^{∫tanx dx} = e^{log secx}
⇒ L.F. = sec x
Applying formula
y × (1.F.) = ∫Q × (I.F.) dx + c
⇒ ∫sec² x dx + c
⇒ y sec x = tan x + c

(B) Solve the differential equation  + y tan x = sin x?
Solution:
Solve as Q.No.1 (A).

Question 2.

Question 3.
Solve the differential equation 3x² dy = (3xy + y²)dx?
Solution:
3x² dy = (3xy + y²) dx (given)

Question 4.
Solve the differential equation (1 + x²)  + 2xy = 4x²?
Solution:

Question 5.
Solve the differential equation (1 + x²)  + 2xy = cos x?
Solution:

Question 6.
Marginal cost price of making anything is given by equation c'(x) =  = 2 + 0.15 x. Find the total cost price c(x) for making it? [Given c (0) = 100]
Solution:
c'(x) =  = 2 + 0.15 x (given)
Integrating both sides
∫c'(x) dx = ∫ (2 + 0.15 x) dx
c(x) = 2x + 0.15  + A …………. (1)
If x = 0
c(0) = 2 × 0 +  × 0² + A
⇒ c(0) = A
∴ A = 100, [∵ c(0) = 100]
Putting in eqn. (1),
c(x) = 2x + 0.075 x² + 100

Question 7.

Question 8.
Solve the differential equation (x + y + 1)  = 1?
Solution:
(x + y + 1)  = 1 (given)
⇒  = x + y + 1
⇒  – x = y + 1
Comparing the equation with  + Px = Q?
P = – 1 and Q = y + 1
∴I.F. = e^∫pdy = e^-∫dy = e^-y
Here the required solution is:
⇒ x.e^∫pdy = ∫e^∫pdy. Qdy
⇒ x.e^-y = ∫e^-y (y + 1) dy + c
⇒ x.e^-y = – (y + 1)e^-y – ∫1(-e^-y) dy + c
⇒ x = -(y + 1) -1 + ce^y
x + y + 2 = ce^y

Question 9.
Solve the differential equation sec²x tany dx + sec² y tan xdy = 0?
Solution:
sec²x tany dx + sec² y tan xdy = 0 (given)

⇒ log y = – log x + log c
⇒ log x + log y = log c
⇒ log xy = log c
⇒ xy = c

Question 10.
Solve the differential equation  = y tanx – 2 sin x?
Solution:
– y tanx = – 2 sin x (given)
Comparing with  + Py = Q,
P = – tan x, Q = – 2 sin x
L.F. = e^∫pdx = e^-∫tanxdx
= e^log_ecosx = cos x
The required solution is:
y.(I.F.) = ∫Q.I.F. dx + c
⇒ ycos x = -2∫sin x cos x dx + c
⇒ ycosx = -∫sin2xdx + c
⇒ ycosx =  + c.

Question 11.
Solve the differential equation  + 2y = 4x?
Solution:
Solution:
+ 2y = 4x
Comparing with  + Py = Q,
P = 2, Q = 4x
I.F. = e^∫pdx = e^∫2dx = e^2x
The required differential solution is:
y. (I.F) = ∫Q.I.F. dx + c
⇒ ye^2x = ∫4xe^2x dx + c

Question 12.
Solve the differential equation cos² x  + y = 2?
Solution:
cos² x  + y = 2
⇒  + sec² x. y = 2 sec² x
Comparing with  + P y = Q,
P = sec² x, Q = 2 sec² x
I.F. = e^∫sec2 xdx 
Required solution is:
y. (I.F) = ∫Q.I.F. dx + c
⇒ y.e^tanx = ∫2 sec² x. e^tanx dx + c
⇒ y.e^tanx = 2.∫e^t dt + c, (Let tan x = t ⇒ sec² xdx = dt)
= 2.e^tanx + c
⇒ e^tanx (y – 2) = c.

Question 13.
Solve the differential equation cos x  + y = sin x?
Solution:
cos x  + y = sin x
⇒  + sec x.y = tan x
Comparing with  + Py = Q,
P = secx, Q = tanx
L.F. = e^∫secxdx
= e^{log_e(sec x + tan x)} dx + c
Required Solution y.I.F. = ∫Q.I.F. dx + c
⇒ y. (sec x + tan x) = ∫tan x.(sec x + tan x) dx + c
= ∫sec x tan x dx + ∫tan² xdx + c
= sec x + ∫(sec² x – 1) dx = sec x + tan x = – x + c.

Question 14.
Solve the differential equation (1 + y²) dx = (tan^-1 y – x) dy? (CBSE 2015)
Solution:
(1 + y²) dx = (tan^-1 y – x) dy (given)

Question 15.
Solve the differential equation (1 + y²) + ( x – e^{tan-1 y})  = 0? (CBSE 2016)
Solution:
The given equation is:

Objective Type Questions

Question 1.
Choose the correct answer:

Question 1.
Degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) does not exist
Answer:
(c) 3

Question 2.
Solution of differential equation (1 + x) y dx + (1 – y) xdy = 0 is:
(a) log xy + x + y = c
(b) logxy + x – y = c
(c) log xy – x – y = c
(d) log xy – x + y = c.
Answer:
(b) logxy + x – y = c

Question 3.
The differential equation of all circles which passes through the origin whose center lie on the A – axis is:
(a) x² = y²+ xy
(b) x² = y² + 3xy
(c) y² = x² + 2xy
(d) y² = x² – 2xy
Answer:
(c) y² = x² + 2xy

Question 4.
Solution of the differential equation  + y = e^x, y(o) = 0 is:
(a) y = e^-x(x – 1)
(b) y = xe^x
(c) y = xe^-x + 1
(d) y = xe^-x
Answer:
(d) y = xe^-x

Question 5.
The straight line which satisfies the differential equation  = m and cuts an intercept 3 on the positive y – axis is:
(a) y = mx + c
(b) y = mx +3
(c) y = mx – 3
(d) y = – mx + 3.
Answer:
(b) y = mx +3

Question 2.
Fill in the blanks:

1. The corresponding differential equation of the equation x² + y² = a² is ……………………………
2. The differential equation of the curve y = e^cx is …………………….. where c is arbitrary constant.
3. The integration factor of the linear differential equation + Py = Q is …………………………….
4. In the linear differential equation  + Py = Q. P and Q are …………………………….
5. The form of differential equation (x + y + 1) dy = dx is ……………………………
6. Solution of the differential equation e^-x+y  = 1 is ……………………………

Answer:

1. y  + x = 0
2. x  = y log y
3. e^∫pdx
4. constant
5. linear differential equation
6. e^-y = e^-x + c

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