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Relations And Functions

Question
CBSEENMA12035790
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Solve the differential equation straight x dy over dx space plus straight y space equals space straight x space cos space straight x space plus space sin space straight x comma space given space that space straight y space equals space 1 space when space straight x space equals space straight x over 2

Solution

The given differential equation is
straight x dy over dx space plus straight y space equals space straight x space cos space straight x space plus space sin space straight x rightwards double arrow space dy over dx space plus straight y over straight x space equals fraction numerator space straight x space cos space straight x space plus space sin space straight x over denominator straight x end fraction
Let space comma space space straight U space equals space 1 over straight x comma space space straight V space space equals space fraction numerator straight x space cos space straight x space plus space sin space straight x over denominator straight x end fraction Now comma space straight I. straight F space equals space straight e integral Udx space equals space space equals space straight e integral 1 over straight x dx space equals space straight e to the power of log space straight x end exponent space equals space straight x Now comma ye integral Udx space equals space integral Ve to the power of integral Udx end exponent dx rightwards double arrow space straight y. straight x space equals space integral fraction numerator straight x space cos space straight x space plus space sin space straight x over denominator straight x end fraction. space straight x space dx rightwards double arrow space xy space equals space integral left parenthesis straight x space cos space straight x space plus space sin space straight x space right parenthesis space dx rightwards double arrow space xy space space equals space integral space straight x space cos space straight x space dx space space plus space integral space sin space straight x space dx rightwards double arrow space xy space equals space space straight x space sin space straight x space minus space integral space sin space straight x space dx space plus space integral sin space straight x space dx space plus space straight C rightwards double arrow space xy space equals space straight x space sin space straight x space plus straight C rightwards double arrow space straight y space equals space sin space straight x space space plus straight C over straight x Given space space colon space straight x space equals space straight pi over 2 space comma space space space straight y space equals 1 rightwards double arrow space 1 space equals space sin space straight pi over 2 space space plus fraction numerator straight C over denominator straight pi divided by 2 end fraction rightwards double arrow space 1 space equals space 1 space plus space fraction numerator straight C over denominator straight pi divided by 2 end fraction rightwards double arrow space straight C thin space equals space 0

Some More Questions From Relations and Functions Chapter

If a matrix has 24 elements, what are the possible orders it can have ? Wh'at. if it has 13 elements ?

lf a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

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