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### Math Assignment Class XII Ch -09 | Differential Equations

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## **Math Assignment Class XII Ch - 09 **

** Differential Equations**

**Extra questions of chapter 09 Differential Equations, class XII with answers and hints to the difficult questions, strictly according to the CBSE Board syllabus. Important and useful math. assignment for the students of class XII****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 – y**

^{2})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**

^{x}dx=(y^{3 }+ 2xe^{y})dy , At y(0) = 1

**Answer: x = - y**

^{2}e^{-y}+ y^{2}/e

**Solution Hint: Given equation can be written as**

**It is Linear Differential Equation in x**

**Integrating Factor = 1/y ^{2}.**

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

**x = - y ^{2}e^{-y} + Cy^{2}.**

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

**So particular solution is: x = - y ^{2}e^{-y} + y^{2}/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)dxAnswer: 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}) sec^{2}y 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**

**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 }= r^{2}**x ^{2 }+ y^{2} = 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**

**(x ^{2 }+ y^{2})y' = 2y(x + yy’)**

**This is the required differential equation**

**Question 22**

**Find the differential equation of the family of curves y ^{2 }= 4ax**

**Ans: 2xyy' - y ^{2} = 0**

**Question 23**

**Find the solution of the differential equation **

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

**2e ^{2y} = x^{4} + C_{1}**

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