Math Assignment Class XII Ch -09 | Differential Equations

Math Assignment | Class XII
Differential Equations|Chapter 09

Extra questions of chapter 09 Differential Equations, class XII  with answers and  hints to the difficult questions, strictly according to the CBSE Board syllabus. 

MATHEMATICS ASSIGNMENT OF EXTRA QUESTION

STRICTLY ACCORDING TO THE PREVIOUS CBSE SAMPLE QUESTION PAPERS AND CBSE BOARD PAPERS

Question: 1 If m and n, respectively, are the order and the degree of the differential equation  

 , then find m + n

Ans: 3

Solution Hint: Solve the above differential we get 

 

Here m = 2 and n = 1 so m + n = 3

Question: 2 Write the order of the differential equation:

Answer: 2

Question: 3 Find the general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0

Answer:  y = C x

Question: 4 Solve the differential equation: ydx + (x – y²)dy = 0

Answer:  

Solution Hint: Reduce it to LDE and then solve it

Question: 5 Solve the differential equation:   

Answer:  

Solution Hint: Reduce it to HDE and then solve it

Question: 6  

Find the general solution of the  differential equation   

Answer:  

Solution Hint: Reduce the given equation into HDE and then solve it.

Question: 7  

Find the particular solution of the following differential equation, given that y = 0 when x = π/4

 

Answer:   

Solution Hint: Convert the given differential equation in LDE

Find Integrating Factor (IF) = sin x

Multiply by integrating factor on both side and then integrating we get 

 

Now putting y = 0 when x = π/4 and find C,  the value of  C is given by

Now require particular solution is 

Question: 8

Write the general solution of differential equation    

Answer:  e^x +e^-y = C

Solution Hint: Given differential equation can be written as 

 

Separating the denominator we get  

Integrating on both side we get: 

⇒ e^x +e^-y = C

Question : 9  Find the particular solution of the following differential equation 

Answer: 

Solution Hint: Use variable separable and integrating on both side we get 

Now putting x = 0 and y = π/4 we get

C = 3√2

Now particular solution is given by

 

Question:10 Find the general solution of the differential equation:

Answer:   xy sin y = sin y - y cos y + C

Solution Hint:

Separating the variables we get LDE shown below

 

IF = y sin y

Multiplying on both side by IF and then integrating we get

x.y sin y = sin y - y cos y + C

Question: 11

Find the particular solution of the differential equation : ye^ydx = (y³ + 2xe^y)dy ,  At  y(0) = 1

Answer: x = - y²e^-y + y²/e

Solution Hint:  Given equation can be written as 

 

It is Linear Differential Equation in x

Integrating Factor = 1/y².

Multiply on both side by I.F. and then integrating we get

x = - y²e^-y + Cy².

Now putting x = 0 and y = 1, we get,  C = 1/e

So particular solution is:  x = - y²e^-y + y²/e

Question 12

Show that (x - y)dy = (x + 2y)dx is a homogenous differential equation. Also, find the general solution of the given differential equation.

Answer:    

Solution Hint:  Given differential equation can be written as 

It is a Homogeneous differential equation. So solving it by making the substitution y/x = v and then integrating we get general solution

 

Question 13

Find the value of (2a - 3b), if a and b represent respectively the order and the degree of the differential equation.

 

Answer: - 5

Order = 2 and Degree = 3

⇒ a = 2 and b = 3

2a - 3b = 2 × 2 - 3 × 3 = 4 - 9 = - 5

Question 14

Solve the differential equation  (e^x + 1)y dy = e^x(y + 1)dx

Answer: log (e^x + 1) = y - log(y + 1) + C

Question 15

Solve the following homogeneous differential equation :  

Answer: y = x ( log |x| + C )

Question 16

Solve the differential equation  

Solution Hint

Write this equation in the standard form given equation reduces to HDE

Now putting y/x = v and integrating on both side we get

log sin v = - log x + log C

Question: 17  Solve the following differential equation   

 

Solution Hint:  Simplify this equation we get

 

This is Linear Differential Equation 

Finding IF and then integrating on both sides we get

  

Question: 18: Find the particular solution of the following differential equation.

e^x tan y dx + (2 - e^x) sec²y dy = 0,  given that y = π/4 when x = 0

Solution Hint: Separate the variables we get 

 

Integrating on both side we get   e^x  – 2 = C tan y

Now putting y = π/4 and x = 0 we get C = -1

Required particular solution is  

y =  tan^-1(2 - e^x )

Question: 19: Solve the followings differential equation

  ,  given  that x = 1 at y = π/2

Solution Hint: 

Reduce this equation into HDE and then putting y/x = v we get 

 

Now integrating on both side and putting v = y/x we get

 

Now putting x = 1 and y = π/2 we get C = 0

Required solution of the given differential equation is  

Question: 20: Solve the following differential equations 

 ,  given that x = 0 at y = 1

Solution Hint: 

Simplify the differential equation we get 

Now integrating on both sides and by using the partial fraction we get 

  

 

When x = 0, y = 1 we get C = 1

Question 21

Find the solution of the differential equation   

Solution Hint:  Separate the variables and the integrating we get: 

2e^2y = x⁴/4 + C₁

PDF form of Assignment

Question 21

Form the differential equation of all circles which is touching the x-axis at the origin.

Solution Hint

Equation of circle with centre C(0, r) and radius r is given by 

(x - 0)² + (y - 0)² = r²

x² + y² = 2ry ........(i)

Differentiating w.r.t. x we get

2x + 2yy' = 2ry'

From this equation find the value of r and putting this value in equation (i) we get

(x² + y²)y' = 2y(x + yy’)

This is the required differential equation

Question 22

Find the differential equation of the family of curves  y² = 4ax

Ans:  2xyy' - y² = 0

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