Question

Find the general solution to \[\dfrac{{dy}}{{dx}} = x{e^y}\] ?

Answer 1

Hint: For the given differential equation, first transform the equation by taking all like variable terms to one side then applying the integration on both sides we will get the required general solution to the given differential equation.

Complete step by step answer:
A differential equation is an equation which includes one or more terms and also includes one variable in terms of the other variable. A general solution of \[{n^{th}}\] order differential equations is defined as the solution that includes \[n\]important arbitrary constants.
Given equation is \[\dfrac{{dy}}{{dx}} = x{e^y}\],
Now take \[dx\] to the right hand side of the equation we get,
\[ \Rightarrow dy = x{e^y}dx\],
Now divide both sides with \[{e^y}\]we get,
\[ \Rightarrow \dfrac{{dy}}{{{e^y}}} = \dfrac{{x{e^y}}}{{{e^y}}}dx\],
Now simplifying we get,
\[ \Rightarrow {e^{ - y}}dy = xdx\],
Now apply integration on both sides we get,
\[ \Rightarrow \int {{e^{ - y}}dy} = \int {xdx} \],
Now applying integration we get,
\[ \Rightarrow - {e^{ - y}} = \dfrac{{{x^2}}}{2} + C\],
Now simplifying we get,
\[ \Rightarrow {e^{ - y}} = C - \dfrac{{{x^2}}}{2}\]
Now applying exponent identity \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\], we get,
\[ \Rightarrow \dfrac{1}{{{e^y}}} = C - \dfrac{{{x^2}}}{2}\],
Now simplifying we get,
\[ \Rightarrow {e^y} = \dfrac{1}{{C - \dfrac{{{x^2}}}{2}}}\],
Now taking L.C.M in the denominator we get,
\[ \Rightarrow {e^y} = \dfrac{2}{{C - {x^2}}}\],
Now taking logarithms on both sides we get,
\[ \Rightarrow \ln {e^y} = \ln \left( {\dfrac{2}{{C - {x^2}}}} \right)\],
Now simplifying we get,
\[ \Rightarrow y = \ln \left( {\dfrac{2}{{C - {x^2}}}} \right)\],
So, the general solution is \[y = \ln \left( {\dfrac{2}{{C - {x^2}}}} \right)\].

\[\therefore \]The general solution for the given differential equation \[\dfrac{{dy}}{{dx}} = x{e^y}\]will be equal to \[y = \ln \left( {\dfrac{2}{{C - {x^2}}}} \right)\].

Note: The general solution of the differential equation is the correlation between the variables \[x\] and \[y\] which is received after removing the derivatives which means applying integration where the relation includes arbitrary constants to represent the order of an equation. The solution of the first-order differential equation includes one arbitrary whereas the second- order differential equation includes two arbitrary constants.