- 2073 Chaitra
Solve dy/dx + y tan x = sec x
Short answer
This is linear with integrating factor sec x. Multiplying and integrating gives y sec x = tan x + c, so y = sin x + c cos x.
Solve: dy/dx + y tan x = sec x.
dy/dx + y tan x = sec x
This is a linear first-order equation, P = tan x, Q = sec x.
Integrating factor
IF = e^(∫tan x dx) = e^(log sec x) = sec x
Solution
y · sec x = ∫ sec x · sec x dx = ∫ sec²x dx = tan x + c
Multiply by cos x:
y = sin x + c cos x
(Check: y' + y tan x = cos x − c sin x + sin x tan x + c sin x = cos x + sin²x/cos x = 1/cos x = sec x ✓.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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