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Find the differential equation whose general solution is the family of curves y = e^x*(a*x + b), where a and b are arbitrary constants.

1. d²y/dx² + 2*dy/dx - y = 0
2. d²y/dx² - 2*dy/dx + y = 0
3. d²y/dx² + 2*dy/dx + y = 0
4. d²y/dx² - 2*dy/dx - y = 0

Correct answer: d²y/dx² - 2*dy/dx + y = 0

Solution

The family e^x*(ax+b) is the solution of an ODE with repeated root m = 1, i.e. (D-1)² y = 0, which expands to y'' - 2y' + y = 0.

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