Question

If $\sin x$ is an integrating factor of the differential equation $\dfrac{{dy}}{{dx}} + {\rm{P}}y = {\rm{Q}}$, write the value of P.

Answer 1

Here, we need to find the value of P. We can obtain the value of P by using the formula for integrating factor and compare it to the given integrating factor $\sin x$. Then, we can simplify the equation to get the value of P.
Formula Used: We have used the formula of integrating factor, $I.F. = {e^{\int {\rm{P}} dx}}$, where P is the function of $x$ and $I.F.$ is the integrating factor.

Complete step by step solution:
The integrating factor is a function that a differential equation can be multiplied by to make the integration of the differential equation simpler. It is given by $I.F.$
We know that the integrating factor of a differential equation of the form $\dfrac{{dy}}{{dx}} + {\rm{P}}y = {\rm{Q}}$ is given by the formula $I.F. = {e^{\int {\rm{P}} dx}}$. Here, P and Q are functions of $x$ and P is a multiple of $y$.
Now, the given integrating factor is $\sin x$.
We will compare $\sin x$ with the formula for an integrating factor to get the value of P.
Comparing $\sin x$ and $I.F. = {e^{\int {\rm{P}} dx}}$, we get
$\sin x = {e^{\int {\rm{P}} dx}}$
Since P is in the exponent of $e$, we take logarithms of both sides to simplify the equation.
$\log \sin x = \log {e^{\int {\rm{P}} dx}}$
We know that $\log {e^x} = x$. Using this property, we can rewrite the equation as
$\log \sin x = \int {\rm{P}} dx$
We know that the derivative of an integral is the function itself. This can be written as $\dfrac{{d\left( {\int {f\left( x \right)} dx} \right)}}{{dx}} = f\left( x \right)$.
Taking the derivative of both sides of the equation $\log \sin x = \int {\rm{P}} dx$, we get
$\begin{array}{l}\dfrac{{d\left( {\log \sin x} \right)}}{{dx}} = \dfrac{{d\left( {\int {\rm{P}} dx} \right)}}{{dx}}\\\\\dfrac{{d\left( {\log \sin x} \right)}}{{dx}} = {\rm{P}}\end{array}$
Thus, the value of P is the derivative of the function $\log \sin x$.
Differentiating the function $\log \sin x$, we get the value of P as
$\begin{array}{l}{\rm{P}} = \dfrac{1}{{\sin x}} \times \dfrac{{d\left( {\sin x} \right)}}{{dx}}\\\ = \dfrac{1}{{\sin x}} \times \cos x\\\ = \dfrac{{\cos x}}{{\sin x}}\\\ = \cot x\end{array}$

$\therefore$ The value of P is $\cot x$.

Note:
Here the important thing that we need to solve the question is the formula for integrating factors. The integrating factor method is used to find the solution of a differential equation. Integrating factors can be usually used to solve linear differential equations of first order. We can make a function integrable by multiplying the differential equation to the differential equation.