# Exercise 45

 Example (i) $\frac{dy}{dx}+\frac{y}{x}=\frac{1}{x^{2}}$ Example (ii) $\frac{dy}{dx}+y\cot{x}=\sec{x}$
 Question 1(i) $\frac{dy}{dx}+\frac{y}{x}=4x^{2}$ Question 1(ii) $2\left(y-4x^{2}\right)+x\frac{dy}{dx}=0$ Question 1(iii) $\frac{dy}{dx}+xy=x$ Question 1(iv) $\left(x^{5}+3y\right)-x\frac{dy}{dx}=0$ Question 1(v) $\frac{dy}{dx}+\frac{2y}{x^{2}-1}=\frac{x^{2}}{x-1}$ Question 1(vi) $\frac{dy}{dx}\cos{x}-y\sin{x}+\cot{x}=0$ Question 1(vii) $\frac{dy}{dx}+y\cot{x}=\sin{2x}$ Question 1(viii) $\frac{dy}{dx}=x^{3}-2xy$ Question 1(ix) $\frac{dy}{dx}=2\left(2x-y\right)$ Question 1(x) $\frac{dy}{dx}-my=ae^{2mx}$ Question 1(xi) $x\ln{x}\frac{dy}{dx}+y=2x^{2}$ Question 1(xii) $x\left(x^{2}+1\right)\frac{dy}{dx}+2y=\left(x^{2}+1\right)^{3}$ Question 1(xiii) $L\frac{di}{dt}+Ri=E$ Question 1(xiv) $\frac{dy}{dx}-my=ae^{mx}$
 Question 2 The population of a country changes at a rate proportional to the current size of the population and the prevailing economic situation. If the economy has a ten-year cycle, resulting in periods of immigration or emigration $\frac{dN}{dt}=kN+c\sin{\left(\frac{\pi t}{5}+a\right)}$ where $N$ is the size of the population at time $t$, measured in years, and $k$, $a$ and $c$ are constants. Find an expression for the size of the population at any time. Question 3 Verify that the differential equation for logistic growth $\frac{dN}{dt}=cN(K-N)$ can be converted into the linear differential equation $\frac{dx}{dt}+cKx=c$ by replacing $N$ by $\frac{1}{x}$. $x=\frac{1}{N}$ $\frac{dx}{dt}=-\frac{1}{N^{2}}\frac{dN}{dt}$ Obtain the equation of logistic growth by first solving the differential equation for $x$ as a function of $t$ and then reintroducing $N$. Question 4 A modification to the logistic equation which accounts for the influence of the resources available on the growth of the population results in the equation $\frac{dN}{dt}=c\frac{\left(K-N\right)N}{K+bN}$ where $b$, $c$ and $K$ are positive constants. Determine the solution of this equation, given that $N=N_0$ when $t=0$.Show that the population size $N$ tends to the carrying capacity of the environment $K$ as $t$ increases.