Please solve 1 and 2 **Problem 1: Differential Equation Solution**1. Solve \( y' = x(y - 3) \).Steps:- Start with \(\frac{dy}{dx} = x(y - 3)\).- Separate the variables: \(dy = x(y - 3)dx\).- Integrate both sides: \(\int \frac{dy}{y - 3} = \int x \, dx\).- The left side becomes \(\ln|y - 3| = \frac{x^2}{2} + C\), where \(C\) is the constant of integration.- Exponentiate to solve for \(y\): \(y - 3 = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}}\).- Finally, express \(y\) as \(y = Ce^{\frac{x^2}{2}} + 3\).**Problem 2: Exponential Growth/Decay Model**2. Pesticide was given to a colony of 8000 ants. If there were 2000 ants left after 2 weeks, find the number of ants after 3 weeks.- The population model is \( y = Ce^{kt} \).**Explanation:**In Problem 1, we solved a differential equation using separation of variables and integration. The resulting function describes a relationship between \(y\) and \(x\).In Problem 2, we are given an exponential decay model where a population of ants is declining over time \(t\). The constants \(C\) and \(k\) need to be determined based on initial conditions and data, typically involving solving for particular values at known time points.

we have been given an equation y'=x(y-3) which can be written as dydx=x(y-3) here, we will use variable separable which says that we have to separate the variables with its respective derivatives and integrate them so, it becomes dyy-3=dx.x∫dyy-3=∫dx.xlog(y-3)=x22+cy-3=ex22cy=ex22c+3we know the general equation of the growth of the ants which is y(t)=cekt (1) it is given that initially there were 8000 ants so, it can be written as y(0)=8000 and there were 2000 left after 2 weeks i.e. y(2)=2000 using these initial solutions we get the value of k and c y(0)=cek(0)8000=c using y(2)=2000 we get y(2)=8000ek(2)2000=8000e2k14=e2kln(14)=2kln e (using ln both sides)k=12ln14 substituting the value of c and k in equation (1) we get y(t)=8000e12ln14twe have to find the number of ants after 3 weeks so, for that we will put t=3 in equation (1) y(3)=8000e12ln14(3)y(3)=8000(0.125)y(3)=1000 so, there will be 1000 ants after 3 weeks