What is an exact differential? Give one example each.
Answers
Exact equation, type of differential equation that can be solved directly without the use of any of the special techniques in the subject. A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation.
Example:-
Example 1.
Solve the differential equation \(2xydx +\) \( \left( {{x^2} + 3{y^2}} \right)dy \) \(= 0.\)
Solution:-
The given equation is exact because the partial derivatives are the same:
\[
{{\frac{{\partial Q}}{{\partial x}} }={ \frac{\partial }{{\partial x}}\left( {{x^2} + 3{y^2}} \right) }={ 2x,\;\;}}\kern-0.3pt
{{\frac{{\partial P}}{{\partial y}} }={ \frac{\partial }{{\partial y}}\left( {2xy} \right) }={ 2x.}}
\]
We have the following system of differential equations to find the function \(u\left( {x,y} \right):\)
\[\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = 2xy\\ \frac{{\partial u}}{{\partial y}} = {x^2} + 3{y^2} \end{array} \right..\]
By integrating the first equation with respect to \(x,\) we obtain
\[{u\left( {x,y} \right) = \int {2xydx} }={ {x^2}y + \varphi \left( y \right).}\]
Substituting this expression for \(u\left( {x,y} \right)\) into the second equation gives us:
\[
{{\frac{{\partial u}}{{\partial y}} }={ \frac{\partial }{{\partial y}}\left[ {{x^2}y + \varphi \left( y \right)} \right] }={ {x^2} + 3{y^2},\;\;}}\Rightarrow
{{{x^2} + \varphi’\left( y \right) }={ {x^2} + 3{y^2},\;\;}}\Rightarrow
{\varphi’\left( y \right) = 3{y^2}.}
\]
By integrating the last equation, we find the unknown function \({\varphi \left( y \right)}:\)
\[\varphi \left( y \right) = \int {3{y^2}dy} = {y^3},\]
so that the general solution of the exact differential equation is given by
\[{x^2}y + {y^3} = C,\]
where \(C\) is an arbitrary constant.
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