Solve each of the following initial value problems:
, y(0) = 1
, y(0) = 1
Given and y(0) = 1
This is a first order linear differential equation of the form
Here, P = tan x and Q = 2x + x2tan x
The integrating factor (I.F) of this differential equation is,
We have
∴ I.F = sec x [∵ elog x = x]
Hence, the solution of the differential equation is,
Recall
⇒ y sec x = x2sec x + c
∴ y = x2 + c cos x
However, when x = 0, we have y = 1
⇒ 1 = 02 + c(cos 0)
⇒ 1 = 0 + c(1)
∴ c = 1
By substituting the value of c in the equation for y, we get
y = x2 + (1)cos x
∴ y = x2 + cos x
Thus, the solution of the given initial value problem is y = x2 + cos x