Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.

4y'' + 8y' + 3y = 0

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THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and distinct, then y(x) = c₁e¹¹x + c₂e¹2x (19) is a general solution of Eq. (16). Thus the solution space of the equation ay" + by' + cy = 0 has basis {ex, e¹2x}.