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.
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