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Important Questions: Differential Equations | Mathematics (Maths) Class 12 - JEE PDF Download

Q1: Integrating factor of the differential equation (1 – x2)dy/dx – xy = 1 is
Ans:
The given differential equation is (1 – x2)dy/dx – xy = 1
(1 – x2)dy/dx = 1 + xy
dy/dx = (1/1 – x2) + (x/1 – x2)y
dy/dx – (x/1 – x2)y = 1/1-x2
This is of the form dy/dx + Py = Q
We can get the integrating factor as below:

Important Questions: Differential Equations | Mathematics (Maths) Class 12 - JEE
Let 1 – x2 = t
Differentiating with respect to x
-2x dx = dt
-x dx = dt/2
Now,
Important Questions: Differential Equations | Mathematics (Maths) Class 12 - JEE
I.F = √t = √(1- x2)

Q2: For each of the given differential equations, find a particular solution satisfying the given condition:
dy/dx = y tan x ; y = 1 when x = 0
Ans:
dy/dx = y tan x
dy/y = tan x dx
Integrating on both sides,

Important Questions: Differential Equations | Mathematics (Maths) Class 12 - JEE
log y = log (sec x) +C
log y = log (C sec x)
⇒ y = C sec x ……..(i)
Now consider y = 1 when x = 0.
1 = C sec 0
1 = C (1)
C = 1
Substituting C = 1 in (i)
y = sec x
Hence, this is the required particular solution of the given differential equation.

Q3: Form the differential equation of the family of circles having a centre on y-axis and radius 3 units.
Ans: 
The general equation of the family of circles having a centre on the y-axis is x2 + (y – b)2 = r2
Given the radius of the circle is 3 units.
The differential; equation of the family of circles having a centre on the y-axis and a radius 3 units is as below:
x2 + (y – b)2 = 32
x2 + (y – b)2 = 9 ……(i)
Differentiating (i) with respect to x,
2x + 2(y – b).y′ = 0
⇒ (y – b). y′ = -x
⇒ (y – b) = -x/y′ …….(ii)
Substituting (ii) in (i),
x2 + (-x/y′)2 = 9
⇒ x2[1 + 1/(y′)2] = 9
⇒ x[(y′)2 + 1) = 9 (y′)2
⇒ (x2 – 9) (y′)2 + x2 = 0
Hence, this is the required differential equation.

Q4: Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.
Ans:
Given,
y = a sin (x + b) … (1)
Differentiating both sides of equation (1) with respect to x,
dy/dx = a cos (x + b) … (2)
Differentiating again on both sides with respect to x,
d2y/dx2 = – a sin (x + b) … (3)
Eliminating a and b from equations (1), (2) and (3),
d2y/ dx2 + y = 0 … (4)
The above equation is free from the arbitrary constants a and b.
This the required differential equation.

Q5: Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation d2y/dx2 + y = 0.
Ans: 
The given function is y = a cos x + b sin x … (1)
Differentiating both sides of equation (1) with respect to x,
dy/dx = – a sinx + b cos x
d2y/dx2 = – a cos x – b sinx
LHS = d2y/dx2 + y
= – a cos x – b sinx + a cos x + b sin x
= 0
= RHS
Hence, the given function is a solution to the given differential equation.

Q6:Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos2(y/x).
Ans
: According to the given condition,
dy/dx = y/x – cos2(y/x) ………….(i)
This is a homogeneous differential equation.
Substituting y = vx in (i),
v + (x) dv/dx = v – cos2v
⇒ (x)dv/dx = – cos2v
⇒ sec2v dv = – dx/x
By integrating on both the sides,
⇒ ∫sec2v dv = – ∫dx/x
⇒ tan v = – log x + c
⇒ tan (y/x) + log x = c ……….(ii)
Substituting x = 1 and y = π/4,
⇒ tan (π/4) + log 1 = c
⇒ 1 + 0 = c
⇒ c = 1
Substituting c = 1 in (ii),
tan (y/x) + log x = 1

Q7: Find the general solution of the differential equation dy/dx = 1 + y2/1 + x2.
Ans: 
Given differential equation is dy/dx = 1 + y2/1 + x2
Since 1 + y2 ≠ 0, therefore by separating the variables, the given differential equation can be written as:
dy/1 + y2 = dx/1 + x…….(i)
Integrating equation (i) on both sides,

Important Questions: Differential Equations | Mathematics (Maths) Class 12 - JEE
tan-1y = tan-1x + C
This is the general solution of the given differential equation.

Q8: Find the differential equation of the family of lines through the origin.
Ans:
Let y = mx be the family of lines through the origin.
Therefore, dy/dx = m
Eliminating m, (substituting m = y/x)
y = (dy/dx) . x
or
x. dy/dx – y = 0

Q9: The number of arbitrary constants in the general solution of a differential equation of fourth order is.
Ans: 
We know that the number of constants in the general solution of a differential equation of order n is equal to its order.
Therefore, the number of constants in the general equation of the fourth-order differential equation is four.
Note: The number of constants in the general solution of a differential equation of order n is equal to zero.

Q10: Determine order and degree (if defined) of differential equation (y′′′)2 + (y″)+ (y′)4 + y5 = 0
Ans:
Given differential equation is (y′′′)2 + (y″)3 + (y′)4 + y= 0
The highest-order derivative present in the differential equation is y′′′.
Therefore, its order is 3.
The given differential equation is a polynomial equation in y′′′, y′′, and y′.
The highest power raised to y′′′ is 2.

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FAQs on Important Questions: Differential Equations - Mathematics (Maths) Class 12 - JEE

1. What is a differential equation?
Ans. A differential equation is a mathematical equation that relates an unknown function to its derivatives. It involves the derivatives of the unknown function with respect to one or more independent variables.
2. What are the applications of differential equations?
Ans. Differential equations have various applications in physics, engineering, economics, biology, and other fields. They are used to model and analyze dynamic systems, such as fluid flow, heat transfer, population dynamics, electrical circuits, and many more.
3. How do you solve a first-order differential equation?
Ans. To solve a first-order differential equation, one can use various methods such as separation of variables, integrating factors, or by transforming it into a linear equation. These methods involve algebraic manipulations and integration to find the solution.
4. What is the order of a differential equation?
Ans. The order of a differential equation is determined by the highest derivative present in the equation. For example, a first-order differential equation contains only the first derivative, while a second-order differential equation contains the second derivative and so on.
5. Can all differential equations be solved analytically?
Ans. No, not all differential equations can be solved analytically. Some differential equations have explicit solutions that can be expressed in terms of elementary functions, while others may require numerical methods or approximation techniques to find an approximate solution. Additionally, some differential equations may not have a closed-form solution at all.
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