Let y = y x be the solution of the differential equation d y = e α x + y d x ; α ∈ N . If y log e 2 = log e 2 and y 0 = log e 1 2 , then the value of α is equal to ___.

Question

Let y = y x be the solution of the differential equation d y = e &#945; x + y d x ; &#945; &#8712; N . If y log e 2 = log e 2 and y 0 = log e 1 2 , then the value of &#945; is equal to ___.

Aicharya · Accepted Answer

We have, &#8747; e - y d y = &#8747; e &#945; x d x &#8658; - e - y = e &#945; x &#945; + c &#160; &#160; &#160; &#160; &#160; &#160; . . . 1 Put x , y = &#8467; n 2 , &#8467; n 2 - 1 2 = 2 &#945; &#945; + c &#160; &#160; &#160; &#160; &#160; &#160; &#160; . . . 2 Put x , y &#8801; 0 , - &#8467; n 2 - 2 = 1 &#945; + c From equation 1 - 2 , we get 2 &#945; - 1 &#945; = 3 2 &#8658; &#945; = 2 ( as &#945; &#8712; N ) .

Let y = y x be the solution of the differential equation d y = e α x + y d x ; α ∈ N . If y…

Solution

Let y = y x be the solution of the differential equation d y = e &#945; x + y d x ; &#945; &#8712; N . If y…

Solution

Let y = y x be the solution of the differential equation d y = e α x + y d x ; α ∈ N . If y…