Find the general solution: \(\frac {dy}{dx}+(sec\ x)y=tan x, \ (0≤x<\frac \pi2)\)
Find the general solution: \(\frac {dy}{dx}+(sec\ x)y=tan x, \ (0≤x<\frac \pi2)\)
Solution and Explanation
The given differential equation is:
\(\frac {dy}{dx}\)+py = Q(where p = sec x and Q = tan x)
Now, I.F = e∫pdx = \(\int\)sec x dx = elog(sec x+tan x) = sec x+tan x
The general solution of the given differential equation is given by the relation,
y(I.F.) = \(\int\)(Q×I.F.)dx + C
⇒y(secx + tanx) = \(\int\)tan x(sec x + tan x)dx +C
⇒y(sec x + tan x) =\(\int\)sec x tan x dx + \(\int\)tan2x dx + C
⇒y(sec x + tan x) = sec x +\(\int\)(sec2x - 1)dx + C
⇒y(sec x + tan x) = sec x + tan x - x + C
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Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
Read More: Formation of a Differential Equation