Question

Find the integrating factor of the differential equation given that $\tan x\dfrac{dy}{dx}+y=2x\tan x+{{x}^{2}}$.

Answer 1

Hint: In this question, first we have to reduce the given differential equation into a linear differential equation of the form$\dfrac{dy}{dx}+Py=Q$. From this equation obtain P and Q. Here, P and Q are functions of x or constants. Then find the integrating factor (I.F.) using the formula $I.F.={{e}^{\int{Pdx}}}$.

Complete step-by-step answer:
The differential equation in this question is,
$\Rightarrow \tan x\dfrac{dy}{dx}+y=2x\tan x+{{x}^{2}}.......(i)$
Now, we have to reduce this equation into a linear equation of the form,
$\Rightarrow \dfrac{dy}{dx}+Py=Q.......(ii)$
This is called a linear differential equation because y, which is the dependent variable and its derivative occurs only in first degree. Integrating factor $I.F.={{e}^{\int{Pdx}}}$ is used to solve this type of equation.
In this question we have to find the integrating factor.
For this equation (i) is divided by $\tan x$. Then equation (i) changes to,
$\begin{align}
& \Rightarrow \dfrac{dy}{dx}+\dfrac{y}{\tan x}=2x+\dfrac{{{x}^{2}}}{\tan x}.......(iii) \\
& \Rightarrow \dfrac{dy}{dx}+y\left( \dfrac{1}{\tan x} \right)=2x+\dfrac{{{x}^{2}}}{\tan x}......(iv) \\
\end{align}$
Now, from equation (iv) we can observe that $P=\dfrac{1}{\tan x}$ and $Q=2x+\dfrac{{{x}^{2}}}{\tan x}$.
Integrating factor for the linear differential equation of the form $\dfrac{dy}{dx}+Py=Q$ is,
$\Rightarrow I.F.={{e}^{\int{Pdx}}}.....(v)$
Now, substituting the values of P in equation (v), we get,
$\begin{align}
& \Rightarrow I.F.={{e}^{\int{\dfrac{1}{\tan x}dx}}} \\
& \Rightarrow I.F.={{e}^{\int{\cot xdx}}}......(vi) \\
\end{align}$
We know that $\int{\cot x=\log |\sin x|}$. Substituting this in equation (vi) we get,
$\Rightarrow I.F.={{e}^{\log |\sin x|}}......(vii)$
$\therefore $ the I.F. of the differential equation $\tan x\dfrac{dy}{dx}+y=2x\tan x+{{x}^{2}}$ is ${{e}^{\log |\sin x|}}$.

Note: The main idea of this problem lies in rearranging the given differential equation into a linear differential equation of the form $\dfrac{dy}{dx}+Py=Q$. Here, P and Q are functions of x or constants. For equations of this form the integrating factor (I.F.) is $I.F.={{e}^{\int{Pdx}}}$.
Please note, if the linear differential equation was of the form $\dfrac{dx}{dy}+Px=Q$, where P and Q are functions of y or constants. For equations of this form the integrating factor (I.F.) is $I.F.={{e}^{\int{Pdy}}}$.