Question

Classify the following differential equation:

exdydx+5y=x3y

1)Linear but not separable

2)Separable but not linear

3)Neither separable nor linear

4)Both separable and linear

Answer 1

${\text{The given differential equation is}} \hfill \\ {e^x}\frac{{dy}}{{dx}} + 5y = {x^3}y \hfill \\ {\text{Multiplying both sides by }}{e^{ - x}}{\text{, we get}} \hfill \\ \frac{{dy}}{{dx}} + 5y{e^{ - x}} = {x^3}y{e^{ - x}} \hfill \\ \Rightarrow \frac{{dy}}{{dx}} + \left( {5 - {x^3}} \right){e^{ - x}}y = 0\,\,\,\,\,\,\,\,\,\left( 1 \right) \hfill \\ {\text{Linear differential equation is given by}} \hfill \\</p><p> \frac{{dy}}{{dx}} + Py = Q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right) \hfill \\</p><p> {\text{where }}P,Q\,{\text{are functions of }}x \hfill \\</p><p> P = \left( {5 - {x^3}} \right){e^{ - x}},Q = 0 \hfill \\</p><p>{\text{Thus, given differential equation is a linear differential equation}}{\text{.}} \hfill \\</p><p>{\text{From (1), we have}} \hfill \\</p><p>\frac{{dy}}{{dx}} + \left( {5 - {x^3}} \right){e^{ - x}}y = 0 \hfill \\</p><p>\Rightarrow \frac{{dy}}{{dx}} = \left( {{x^3} - 5} \right){e^{ - x}}y \hfill \\</p><p>\Rightarrow \frac{{dy}}{y} = \left( {{x^3} - 5} \right){e^{ - x}}dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 3 \right) \hfill \\</p><p>{\text{Thus, given differential equation is separaable also}}{\text{.}} \hfill \\</p><p>{\text{Hence, the given differential equation is linear as well as separable}}{\text{.}} \hfill \\</p><p>$