For each of the following systems of linear differential...

For each of the following systems of linear differential equation systems, solve using: (a) the substitution method (b) the direct method Ensure that the solutions satisfy any initial conditions that are given. (i) $\dot{y}_1=y_1+5 y_2, \dot{y}_2=\frac{1}{4} y_1-y_2$ (ii) $\dot{y}_1=y_1+5 y_2+18, \dot{y}_2=\frac{1}{4} y_1-y_2+9$, and $y_1(0)=6, y_2(0)=0$ (iii) $\dot{y}_1=2 y_1+y_2 / 2, \dot{y}_2=7 y_1 / 2-y_2+15$, and $y_1(0)=2, y_2(0)=4$ (iv) $\dot{y}_1=3 y_1+y_2+4, \dot{y}_2=2 y_1+2 y_2-12$, and $y_1(0)=-2, y_2(0)=5$

For each of the following systems of linear differential equation systems, solve using:
(a) the substitution method
(b) the direct method

Ensure that the solutions satisfy any initial conditions that are given.
(i) $\dot{y}_1=y_1+5 y_2, \dot{y}_2=\frac{1}{4} y_1-y_2$
(ii) $\dot{y}_1=y_1+5 y_2+18, \dot{y}_2=\frac{1}{4} y_1-y_2+9$, and $y_1(0)=6, y_2(0)=0$
(iii) $\dot{y}_1=2 y_1+y_2 / 2, \dot{y}_2=7 y_1 / 2-y_2+15$, and $y_1(0)=2, y_2(0)=4$
(iv) $\dot{y}_1=3 y_1+y_2+4, \dot{y}_2=2 y_1+2 y_2-12$, and $y_1(0)=-2, y_2(0)=5$

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