question_answer

\[(x-1)\,dy+y\,dx=x(x-1){{y}^{1/3}}dx,\]where x denotes the percentage of population living in a city and y denotes the area for living a healthy life of population. Find the particular solution, when x = 2 and y = 1. Is higher density of population is harmful? Justify your answer.

Answer:

We have, \[(x-1)\,dy+y\,dx=x(x-1){{y}^{1/3}}dx\]   On dividing both sides by \[dx\,\,{{y}^{1/3}}(x-1)dx,\] the given Equation reduces to \[{{y}^{-1/3}}\frac{dy}{dx}+\frac{1}{x-1}{{y}^{2/3}}=x\]  Put \[{{y}^{2/3}}=V\] \[\Rightarrow \] \[\frac{2}{3}{{y}^{-1/3}}\frac{dy}{dx}=\frac{dV}{dx}\] Then, given equation reduces to             \[\frac{dV}{dx}+\frac{2}{3(x-1)}V=\frac{2}{3}x\] which is a linear differential equation of the form \[\frac{dV}{dx}+PV=Q.\] Here, \[P=\frac{2}{3(x-1)}\] and \[Q=\frac{2}{3}x\] \[\therefore \] IF \[={{e}^{\int{Pdx}}}={{e}^{\frac{2}{3}\int{\frac{1}{(x-1)}dx}}}={{e}^{\frac{2}{3}\log (x-1)}}={{(x-1)}^{2/3}}\] Hence, the solution is given by             \[V{{(x-1)}^{2/3}}=\frac{2}{3}{{\int{x(x-1)}}^{2/3}}dx+C\]   Put \[x-1={{t}^{3}}\] \[\Rightarrow \] \[dx=3{{t}^{2}}dt\] in the RHS, we get \[\int{x}{{(x-1)}^{2/3}}dx=\int{({{t}^{3}}+1)}{{t}^{2}}\cdot 3{{t}^{2}}dt=3\int{({{t}^{7}}+{{t}^{4}})dt}\] \[=3\left[ \frac{1}{8}{{t}^{8}}+\frac{1}{5}{{t}^{5}} \right]=\frac{3}{8}{{(x-1)}^{8/3}}+\frac{3}{5}{{(x-1)}^{5/3}}\] Hence, the required solution is \[{{y}^{2/3}}{{(x-1)}^{2/3}}=\frac{2}{3}\left[ \frac{3}{8}{{(x-1)}^{8/3}}+\frac{3}{5}{{(x-1)}^{5/3}} \right]+C\] \[\Rightarrow \] \[{{y}^{2/3}}=\frac{1}{4}{{(x-1)}^{2}}+\frac{2}{5}(x-1)+C{{(x-1)}^{-2/3}}\]  ? (i) When x = 2 and y = 1, then \[{{(1)}^{2/3}}=\frac{1}{4}{{(2-1)}^{2}}+\frac{2}{5}(2-1)+C{{(2-1)}^{-2/3}}\] \[\Rightarrow \]\[1=\frac{1}{4}+\frac{2}{5}+C\] \[\Rightarrow \] \[C=1-\frac{1}{4}-\frac{2}{5}=\frac{20-5-8}{20}=\frac{7}{20}\] On putting \[C=\frac{7}{20}\] in Eq. (i), we get \[{{y}^{2/3}}=\frac{1}{4}{{(x-1)}^{2}}+\frac{2}{5}(x-1)+\frac{7}{20}{{(x-1)}^{-2/3}},\]which is the required particular solution of the given differential equation. Value Yes, higher density of population is very harmful for human beings. Some major impacts of high population are as follows:        (a) Unemployment (b) Pressure on infrastructure (c) Decreased production and increased costs.