Question

Find the integrating factor of the differential equation $\left( {\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}\, - \,\dfrac{y}{{\sqrt x }}} \right)\,\dfrac{{dx}}{{dy}} = 1$

Answer 1

The given differential is not given in any of the standard forms. Since integrating factor has to be calculated, we need to express the given differential equation in one of the standard forms which are as follows:
$\dfrac{{dy}}{{dx}} + P(x)y = \,Q(x)$ or,
$\dfrac{{dx}}{{dy}} + P(y)c = Q(y)$
After expressing the differential equation in standard form, we obtain the expression of P.We can then use the formula for integrating factor ${e^{\int {Pdx} }}$ or ${e^{\int {Pdy} }}$
Given:
The differential equation is given as $\left( {\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}\, - \,\dfrac{y}{{\sqrt x }}} \right)\,\dfrac{{dx}}{{dy}} = 1$

Complete step by step solution:
Since, $\left( {\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}\, - \,\dfrac{y}{{\sqrt x }}} \right)\,\dfrac{{dx}}{{dy}} = 1$
$\Rightarrow \,\dfrac{{dy}}{{dx}} = \,\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \dfrac{y}{{\sqrt x }}$
$\Rightarrow \dfrac{{dy}}{{dx}} + \dfrac{y}{{\sqrt x }} = \,\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}$
The above differential equation is of the form
$\dfrac{{dy}}{{dx}} + \,P(x)y = Q(x)$
Therefore, the integrating factor is given by
Integrating Factor=${e^{\int {P(x)dx} }}$
On comparing the differential equation with the standard form, we have
$P(x) = \dfrac{1}{{\sqrt x }}$
Therefore, Integrating Factor = ${e^{\int {\dfrac{1}{{\sqrt x }}dx} }}$
We know that,
$\int {\dfrac{{dx}}{{\sqrt x }}} \, = 2\sqrt x$
Therefore, Integrating Factor= ${e^{2\sqrt x }}$
Integrating Factor is ${e^{2\sqrt x }}$.
Thus, the integrating factor of the differential equation
$\left( {\dfrac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}\, - \,\dfrac{y}{{\sqrt x }}} \right)\,\dfrac{{dy}}{{dx}} = 1$ is ${e^{2\sqrt x }}$.

Note:
The two standard forms of the differential equation mentioned in the hint are extremely important. The students must keep in mind both the forms and should practice enough to identify and then reduce the differential equation in these forms whenever it is possible. This form of differential equation is helpful in solving a special type of differential equation which is called Bernoulli differential equation.