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- 2072 Chaitra
Solve x log x dy/dx + y = 2 log x
Short answer
Divide by x log x to get linear form with integrating factor log x. Multiplying and integrating (log x)² gives the solution y = log x + c/log x.
Solve the differential equation: x log x dy/dx + y = 2 log x.
x log x dy/dx + y = 2 log x
Standard linear form
Divide by x log x:
dy/dx + y/(x log x) = 2/x
P = 1/(x log x), Q = 2/x.
Integrating factor
∫ dx/(x log x) = log(log x), so IF = log x.
Solution
y log x = ∫ (2/x) log x dx = (log x)² + c
(since ∫ (log x)/x dx = (log x)²/2)
y = log x + c/log x
(Check: y' = 1/x − c/(x (log x)²); x log x · y' = log x − c/log x; adding y = log x + c/log x gives 2 log x ✓.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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