# 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 [math]N[/math] is the size of the population at time [math]t[/math], measured in years, and [math]k[/math], [math]a[/math] and [math]c[/math] are constants. |

Question 3 | Verify that the differential equation for logistic growth [math]\frac{dN}{dt}=cN(K-N)[/math] can be converted into the linear differential equation [math]\frac{dx}{dt}+cKx=c[/math] by replacing [math]N[/math] by [math]\frac{1}{x}[/math].
\[x=\frac{1}{N}\] \[\frac{dx}{dt}=-\frac{1}{N^{2}}\frac{dN}{dt}\] Obtain the equation of logistic growth by first solving the differential equation for [math]x[/math] as a function of [math]t[/math] and then reintroducing [math]N[/math]. |

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 [math]b[/math], [math]c[/math] and [math]K[/math] are positive constants. Determine the solution of this equation, given that [math]N=N_0[/math] when [math]t=0[/math]. |