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Solve x dy/dx + y log y = xy eˣ
Short answer
Divide by xy, set z=log y to obtain dz/dx + z/x = eˣ. Integrating factor x gives xz = (x−1)eˣ + c. Hence x log y = (x−1)eˣ + c.
Solve the differential equation x dy/dx + y log y = xy eˣ.
x dy/dx + y log y = xy eˣ
Reduce to a linear equation
Divide by xy:
(1/y) dy/dx + (log y)/x = eˣ
Put z = log y, so dz/dx = (1/y) dy/dx:
dz/dx + z/x = eˣ
This is linear in z with P = 1/x, Q = eˣ.
Integrating factor
IF = e^(∫dx/x) = x
Solution
z · x = ∫ x eˣ dx = x eˣ − eˣ + c = (x − 1)eˣ + c
Replace z = log y:
x log y = (x − 1)eˣ + c, i.e. log y = (1 − 1/x)eˣ + c/x.
✓ Answer checked by Claude Opus 5.5 (high effort)
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