2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise

   

Students can Download Maths Chapter 9 Differential Equations Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise

Question 1.
For each of the differential equations given below, indicate its order and degree
(if defined).

(i)
\(\cfrac{d^{2} y}{d x^{2}}+5 x\left(\cfrac{d y}{d x}\right)^{2}-6 y=\log x\)
Answer:
Order -2, degree – 1

(ii)
\(\left(\cfrac{d y}{d x}\right)^{3}-4\left(\cfrac{d y}{d x}\right)^{2}+7 y=\sin x\)
Answer:
Order is 1 degree is 3.

(iii)
\(\cfrac{d^{4} y}{d x^{4}}-\sin \left(\cfrac{d^{3} y}{d x^{3}}\right)=0\)
Answer:
Order is 4. As the equation is not a polynomial in \(\frac{d y}{d x}\), degree is not defined.

KSEEB Solutions

Question 2.
For each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i) y = a ex + b e-x + x2
\(x \cfrac{d^{2} y}{d x^{2}}+2 \cfrac{d y}{d x}-x y+x^{2}-2=0\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 1

(ii) y = ex (a cos x + b sin x)
\(\cfrac{d^{2} y}{d x^{2}}-2 \cfrac{d y}{d x}+2 y=0\)
Answer:
\(\frac{d y}{d x}\) = ex (-a sin x + b cos x) + (+a cos x + b sin x) ex
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 2

(iii) \(y=x \sin 3 x \quad: \cfrac{d^{2} y}{d x^{2}}+9 y-6 \cos 3 x=0\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 3

KSEEB Solutions

(iv) \(2 x=2 y^{2} \times \cfrac{1}{y} \times \cfrac{d y}{d x}+\log y \times 4 y \times \cfrac{d y}{d x}\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 4

Question 3.
Form the differential equation representing the family of curves given by
(x – a)2 + 2y2 = a2, where a is an arbitrary constant.
Answer:
(x – a)2 + 2y2 = a2
2(x -a ) +2(2y) x \(\frac{d y}{d x}\) = 0
(x – a) + 2yx y’ = 0, 2y y’ = – (x – a) = a – x ………….(1)
4y2 (y’ )2 + 2 y2 = 4y2 (y’ )2 + 4 xy y’ + x2
\(4 x y \frac{d y}{d x}+x^{2}-2 y^{2}=0\)

Question 4.
Prove that x2 – y2 = c (x2 + y2) is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.
Answer:
x2 – y2 = c (x2 + y2)2
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 5

KSEEB Solutions

Question 5.
Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Answer:
(x – h)2 + (y – h)2 = h2 dy
2(x – h) +2(y – h) \(\frac{d y}{d x}=0\)
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 6
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 7

Question 6.
Find the general solution of the differential equation
\(\cfrac{d y}{d x}+\sqrt{\cfrac{1-y^{2}}{1-x^{2}}}=0\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 8

Question 7.
Show that the general solution of the different equation
\(\frac{d y}{d x}+\frac{y^{2}+y+1}{x^{2}+x+1}=0\) is given by
(x +y+1) = A (1- x – y -2xy)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 9
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 10

KSEEB Solutions

Question 8.
Find the equation of the curve passing through the point \(\left(0, \frac{\pi}{4}\right)\) whose differencial  equation is sin x y dy = 0
Answer:
Sin x cos y dx = – cos x sin y dy
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 11

Question 9.
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 12

Question 10.
Solve the differential equation
\(\mathbf{y}^{\mathbf{e}^{\frac{x}{y}}} \mathbf{d x}=\left(\mathbf{x e}^{\frac{\mathbf{x}}{y}}+\mathbf{y}^{2}\right) \mathbf{d y}(\mathbf{y} \neq \mathbf{0})\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 13

Question 11.
Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = -1, when x = 0. (Hint: put x – y = t)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 44
x + y = log (x – y) + C
when x = 0, y = -1
0 – 1 = log(0 + 1) + C
⇒ C = -1 solution is x + y = log (x – y) – 1.

KSEEB Solutions

Question 12.
Solve the differential equation
\(\left[\cfrac{\mathrm{e}^{-2 \sqrt{x}}}{\sqrt{x}}-\cfrac{y}{\sqrt{x}}\right] \cfrac{d x}{d y}=1(x \neq 0)\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 14

Question 13.
Find a particular solution of the differential dy equation \(\frac{d y}{d x}+y \cot x=4 x \csc x(x \neq 0)\) given that y = 0 when \(x=\frac{\pi}{2} \)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 15

Question 14.
Find a particular solution of the differential dy equation (x+1) \frac{d y}{d x}=2 e^{-y}-1, given that y = 0 when x = 0.
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 16

KSEEB Solutions

Question 15.
The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 17
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 18
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 19

Question 16.
The general solution of the differential equation \(\frac{y d x-x d y}{y}=0\) is
(A) xy = C
(B) x = Cy2
(C) y = Cx
(D) y = Cx2
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 20

Question 17.
The general solution of a differential equation of the type \(\frac{d x}{d y}+P_{1} x=Q_{1} \text { is }\)
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 21
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 22

KSEEB Solutions

Question 18.
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
(A) x ey+ x2 = C
(B) x ey + y2 = C
(C) y ex + x2 = C
(D) y ey + x2 = C
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 23

2nd PUC Maths Differential Equations Miscellaneous Exercise Additional Questions and Answers

Question 1.
Solve the differential equations
\(\sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \cfrac{d y}{d x}=0\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 24

Question 2.
solve
\(\left(x^{2}-1\right) \cfrac{d y}{d x}+2 x y=\cfrac{1}{x^{2}-1}\)(CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 25
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 26

Question 3.
Show that \((x-y) \frac{d y}{d x}=x+2 y\) is homogenous and solve it (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 27
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 28

KSEEB Solutions

Question 4.
Solve
\(y d x+x \log \left(\cfrac{y}{x}\right) d y-2 x d y=0\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 29
Question 5.
solve
\(\left(x^{3}+x^{2}+x+1\right)\left(\cfrac{d y}{d x}\right)=2 x^{2}+x\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 30

Question 6.
Solve
\(\left(x^{2}+1\right) \cfrac{d y}{d x}+2 x y=\sqrt{x^{2}+4}\)(CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 31
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 32

Question 7.
Solve \(x d y-y d x=\sqrt{x^{2}+y^{2}} d x\)(CBSE 2011)
Answer:

2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 33

KSEEB Solutions

Question 8.
Solve
\(\left(y+3 x^{2}\right) \frac{d x}{d y}=x\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 34
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 35

Question 9.
Solve x dy + (y – x3) dx = 0 (CBSE 2011)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 43

Question 10.
Form the differential equation of the firmly of circles in the second quadrant and touching the co-ordinate axes. (CBSE 2012)
Answer:
Centre is (-h, h), v = h
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 36
equation of circle is (x + h)2 + (y – h)2 = h2
(x – h)2 + (y – h)2 = h2 dy
2(x – h) +2(y – h) \(\frac{d y}{d x}=0\)
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 37
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 38

KSEEB Solutions

Question 11.
(1 – x)2 dy + 2 xy dx = cot x dx
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 39

Question 12.
\(\int x\left(x^{2}-1\right) \frac{d x}{d y}=1, y=0, \text { when } x=2\) (CBSE 2012)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 40

Question 13.
Find the particular solution: (CBSE 2012)
\(x \cfrac{d y}{d x}-y+x \sin \left(\cfrac{y}{x}\right)=0, \text { when } x=2, y=\pi\)
Answer:
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 41
2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise 42

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