Question 3.3.1: Solve the following differential equations. (a) 2y" - 5y' - ......

We give the auxiliary equations, the roots, and the corresponding general solutions.

(a) 2 m^2-5 m-3=(2 m+1)(m-3), m_1=-\frac{1}{2}, m_2=3 \text {. From (4), }

y=c_1 e^{-x / 2}+c_2 e^{3 x} .

(b) m^2-10 m+25=(m-5)^2, m_1=m_2=5 \text {. From (6), }

y=c_1 e^{5 x}+c_2 x e^{5 x} .

(c) m^2+4 m+7=0, m_1=-2+\sqrt{3} i, m_2=-2-\sqrt{3} i \text {. From (8) with } \alpha=-2, \beta=\sqrt{3} , we have

y=e^{-2 x}\left(c_1 \cos \sqrt{3} x+c_2 \sin \sqrt{3} x\right) .