Question

Let y(x) be the solution of the differential equation $left parenthesis straight x space log space straight x right parenthesis dy over dx space plus straight y space equals space 2 space straight x space log space straight x comma$ (x ≥1). Then, y (e) is equal to

e

0

2

2e

Solution

Given differential equation is
$straight x left parenthesis log space straight x right parenthesis dy over dx space plus straight y space equals 2 space straight x space log space straight x comma rightwards double arrow space dy over dx space plus fraction numerator straight y over denominator straight x space log space straight x end fraction space equals 2$
This is a linear differential equation.
therefore, $IF space equals space straight e to the power of integral fraction numerator 1 over denominator straight x space log space straight x end fraction dx end exponent space equals space straight e to the power of log space left parenthesis log space straight x right parenthesis space end exponent space equals space log space straight x$
Now, the solution of given differential equation is given by
$straight y. space log space straight x space equals space integral log space straight x. space 2 dx straight y. space log space straight x space equals space space equals space 2 integral log space straight x space dx At space straight y. space log space straight x space equals space 2 left square bracket space straight x space log space straight x minus straight x right square bracket plus straight c straight x space equals space 1 straight c space equals 2 straight y. space log space straight x space equals space 2 left square bracket straight x space log space straight x minus straight x right square bracket plus 2 straight x space equals straight e straight y space equals space 2 space left parenthesis straight e minus straight e right parenthesis space plus 2 space rightwards double arrow straight y space equals space 2$

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Let y(x) be the solution of the differential equation $left parenthesis straight x space log space straight x right parenthesis dy over dx space plus straight y space equals space 2 space straight x space log space straight x comma$ (x ≥1). Then, y (e) is equal to

e

0

2

2e

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Let y(x) be the solution of the differential equation (x ≥1). Then, y (e) is equal to e 0 2 2e

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:y' + y = ex

Determine order and degree (if defined) of differential equations:y'' + (y')2 + 2 y = 0

Determine order and degree (if defined) of differential equations:y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Let y(x) be the solution of the differential equation $left parenthesis straight x space log space straight x right parenthesis dy over dx space plus straight y space equals space 2 space straight x space log space straight x comma$ (x ≥1). Then, y (e) is equal to

e

0

2

2e

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0