Exam (elaborations)
QUESTION BANK mathematics (DIFFERENTIAL EQUATIONS)
- Course
- Institution
THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER DIFFERENTIAL EQUATIONS ALONG WITH THE ANSWER KEY AT THE END
[Show more]
Seller
Follow
yashashmathur
Content preview
Differential EquationsQUICK RECAP
DIFFERENTIAL EQUATION X A differential equation involving derivatives
of the dependent variable with respect to
8 An equation involving an independent variable, only one independent variable is called an
a dependent variable and the derivatives of ordinary differential equation.
the dependent variable is called differential X A differential equation involving derivatives
equation. with respect to more than one independent
, variables is called a partial differential FORMATION OF A DIFFERENTIAL
equation. EQUATION
8 Order and Degree of a Differential 8 Differential equation of the equation of a
Equation family of curves is obtained by eliminating
X The order of highest derivative appearing in arbitrary constants occurring in it with the
a differential equation is called order of the help of equation of the curves and equations
differential equation. formed by its differentiation.
X The power of the highest order derivative
appearing in a differential equation, after it METHODS OF SOLVING DIFFERENTIAL
is made free from radicals and fractions, is EQUATIONS
called degree of the differential equation.
8 Equation in Variable Separable Form : If
Note : Order and degree (if defined) of a
the differential equation is of the form
differential equation are always positive
integers. f (x) dx = g(y) dy, then the variables are
separable and such equations can be solved
HOMOGENEOUS DIFFERENTIAL EQUATIONS by integrating on both sides. The solution is
8 A differential equation of the form given by
dy f (x , y ) ∫ f (x) dx = ∫ g(y) dy + C, where C is an arbitrary
= constant.
dx g (x , y )
where, f (x, y) and g (x, y) are homogeneous 8 Equation Reducible to Homogeneous Form :
functions of x and y of the same degree. If the equation is of the form dy = f (x , y ) ,
dx g (x , y )
LINEAR DIFFERENTIAL EQUATIONS
where f(x, y) and g(x, y) are homogeneous
dy functions of the same degree in x and y, then
8 An equation of the form + Py = Q where
dx dy dv
put y = vx and =v+x so that the
P and Q are functions of x only (or constants) dx dx
is called a linear differential equation of the dependent variable y is changed to another
first order. variable v, then apply variable separable
SOLUTION OF A DIFFERENTIAL EQUATION method.
8 Solution of a differential equation is a 8 Solution of Linear Differential Equation :
function of the form y = f (x) + C which A differential equation of the form
satisfies the given differential equation. dy
+ Py = Q , where P and Q are functions
dx
8 General Solution : The solution of a
differential equation which contains a of x (or constants) can be solved as :
number of arbitrary constants equal to the
Find Integrating Factor (I.F.) = e ∫
P dx
1.
order of the differential equation.
2. The solution of the differential equation
8 Particular Solution : A solution obtained
is y(I.F.) = ∫ Q (I.F.) dx + C , where C is
by giving particular values to arbitrary
constants in the general solution. constant of integration.
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
9.2 Basic Concepts 10. Write the degree of the differential equation
2
d 2 s ds 3
VSA (1 mark) 2 + + 4 = 0. (AI 2013C)
dt dt
1. The degree of the differential equation
dy
2 9.3 General and Particular
1 + = x is . (2020)
dx Solutions of a Differential
2. Find the order and the degree of the differential Equation
4
dy 2
2 d2 y VSA (1 mark)
equation x = 1 + .
dx2 dx 11. The number of arbitrary constants in the
(Delhi 2019) particular solution of a differential equation
of second order is (are)
3. Write the sum of the order and degree of the
(a) 0 (b) 1
following differential equation
(c) 2 (d) 3 (2020)
d dy
3
= 0. (AI 2015)
dx dx 9.4 Formation of a Differential
4. Write the sum of the order and Equation whose General
degree of the differential equation Solution is given
2
d 2 y dy 3
2 + + x 4 = 0 .(Foreign 2015) VSA (1 mark)
dx dx
12. Form the differential equation representing
5. Write the sum of the order and degree of the A
differential equation the family of curves y = + 5 , by
x
4 3 eliminating the arbitrary constant A.
dy d2 y
1+ = 7 2 . (Delhi 2015C) (AI 2019)
dx dx
13. Find the differential equation representing
6. Write the degree of the differential equation A
4 the family of curves v = + B, where A and
dy d2 y r
+ 3x 2 = 0. (Delhi 2013) B are arbitrary constants. (Delhi 2015)
dx dx
14. Write the differential equation obtained
7. Write the degree of differential equation by eliminating the arbitrary constant C
2 4
2
3 d y dy in the equation representing the family of
x 2 + x = 0. (Delhi 2013) curves xy = C cos x. (Delhi 2015C)
dx dx
15. Write the differential equation representing
8. Write the degree of the differential equation the family of curves y = mx, where m is an
3 4
d2 y dy arbitrary constant. (AI 2013)
x 2 + y + x 3 = 0 . (Delhi 2013)
dx dx SA (2 marks)
9. Write the degree of the differential equation :
16. Form the differential equation representing
4
dy d2 y the family of curves y = e2x(a + bx), where ‘a’
+ 3 y 2 = 0 (Delhi 2013C)
and ‘b’ are arbitrary constants. (Delhi 2019)
dx dx
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller yashashmathur. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67096 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now