\(y +c_a = 23t + c_b\)                              With \(c_b - c_a = c\)
\(\dot{y}=23\)
Using the conditions:
\(1=23*0+c\)
\(c=1\)
\(y=23t+1\)
3) \(\dot{y}+3y=2\)
\(y(0)=4\)
\(\dot{y}+3y=0\)
\(\frac{dy}{dt}=-3y\)
\(dy=-3y dt\)
\(\frac{1}{y}dy=-3dt\)
\(\int\frac{1}{y}dy= -3\int dt\)
\(ln(y) + c_a = -3t +c_b\)                          With \(c_b - c_a = c\)
\(y=e^{-3t+c}\)                                                        With \(A=e^c\)
\(y=Ae^{-3t}\)
\(\dot{y}=-3Ae^{-3t}\)
Using the conditions:
\(-3Ae^{-3*0} + 3*4=2\)
\(A = \frac{10}{3}\)
\(y= \frac{2}{3}-\frac{\dot{y}}{3}\)
\(y= \frac{2}{3}-\frac{1}{3} [-3\frac{10}{3}e^{-3t}]\)
\(y= \frac{2}{3}+\frac{10}{3}e^{-3t}\)
4) \(\dot{y} - 7y = 7\)
\(y(0)=7 \)
\(\dot{y}-7y=0\)
\(\frac{dy}{dt}=7y\)
\(\frac{1}{y} dy = 7dt\)
\(\int \frac{1}{y} dy = 7 \int dt\)
\(ln(y) + c_a = 7t + c_b\)                            With   \(C = c_b - c_a\)
\(e^{ln(y)} = e^{7t}e^C\)                                             \(A=e^C\)
\(y=Ae^{7t}\)
\(\dot{y}=7Ae^{7t}\)
Using the conditions:
\(7Ae^{7*0} - 49 = 7\)
\(A=8\)
\(-7y=7-\dot{y}\)
\(y=-1+\frac{\dot{y}}{7}\)
\(y=8e^{7t} -1\)
5) \(3\dot{y}+6y=5\)
\(y(0)=0\)
\(3\dot{y}+6y=0\)
\(3\dot{y}=-6y\)
\(\frac{dy}{dt}=-2y\)
\(dy=-2ydt\)
\(\int \frac{1}{y} dy = -2 \int dt\)
\(ln(y) + c_a = -2t +c_b\)                             With \(C=c_b-c_a\)
\(e^{ln(y)}=e^{-2t}e^C\)                                                With \(A= e^C\)
\(y=Ae^{-2t}\)
\(\dot{y}=-2Ae^{-2t}\)
Using the conditions:
\(-6Ae^{-2*0} = 5\)
\(A= \frac{-5}{6}\)
\(6y=5-3\dot{y}\)
\(y= \frac{5}{6} - \frac{5}{6}e^{-2t}\)

Find the general solution and draw the phase diagram for the following linear system:

6) \([\begin{matrix} \dot{x}_1 \\ \dot{x}_2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} x_1 \\ x_2 \end{matrix}]\) 
\(\dot{x}_1 =x_1\)
\(\dot{x}_2 =x_2\)
\(\frac{dx_1}{dt}=x_1\)
\(dx_1=x_1dt\)
\(\int \frac{1}{x_1}dx_1= \int dt\)
\(ln(x_1) +c_a = t+c_b\)
\(e^{ln(x_1)} = e^t e^{c_1}\)
\(x_1 = A_1e^t\)
\(\frac{dx_2}{dt}=x_2\)
\(dx_2=x_2dt\)
\(\int \frac{1}{x_2}dx_2= \int dt\)
\(ln(x_2) +c_c = t+c_d\)
\(e^{ln(x_2)} = e^t e^{c_2}\)
\(x_2 = A_2e^t\)
\([\begin{matrix} x_1 \\ x_2 \end{matrix}] = [\begin{matrix} A_1e^t \\ A_2e^t \end{matrix}]\)