6. Given the general income model ( no government sector):
\[
\begin{array}{l}
\mathrm{Y}=\mathrm{C}+\mathrm{I} \ldots \ldots \\
\mathrm{C}=\mathrm{C}_{0}+\mathrm{bY}
\end{array}
\]
Where \( \mathrm{I}=\mathrm{I}_{0}, 0<\mathrm{b}<1 \) ( \( \mathrm{b}, \mathrm{I}_{0} \), and \( \mathrm{C}_{0} \) are constants)
(a) Write equations 1 and 2 in the form : \( a_{1} Y+a_{2} C=a_{3} \quad a_{1}, a_{2}, a_{3} \) are constants
(b) Hence use Cramer's rule to express the equilibrium levels of income and consumption in terms of constants b, \( \mathrm{I}_{0} \), and \( \mathrm{C}_{0} \).
7. Find the particular solution to the following differential equation that satisfies the given initial condition(s).
a) \( \frac{d y}{d x}=\frac{\left(y^{2}+1\right) x}{y} \quad \mathrm{y}(0)=-1 \)
b) \( \ddot{y}-6 \dot{y}+9 y=0 \quad \mathrm{y}(0)=1 \) and \( j(0)=1 \)
c) \( \frac{d y}{d x}=\frac{x y}{x^{2}+1} \)
\[
y(0)=2
\]
d) \( y=e^{\text {s+ }}, \quad \mathrm{y}(0)=0 \)
8. A population of weevils is growing at rate of \( 3 \% \) per year. Let \( w_{n} \) be the number of weevils \( n \) years from now and suppose that there are currently 350 weevils.
(a) Write a difference equation which describes how the population changes from year to year.
(b) Solve the difference equation of part (a).
(c) If the population growth continues at the rate of \( 3 \% \), how many weevils will there be 15 years from now?
