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

Hint:
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.