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Question: Find the general solution of the differential equation \[\dfrac{{dy}}{{dx}} - y = \sin x\]...

Find the general solution of the differential equation
dydx−y=sin⁡x\dfrac{{dy}}{{dx}} - y = \sin x

Explanation

Solution

Hint: The given differential equation is of the form:
dydx+p(x)y=Q(x)\dfrac{{dy}}{{dx}} + p\left( x \right)y = Q\left( x \right)
Where P(x)P\left( x \right) and Q(x)Q\left( x \right) are functions of x only or constants. To find the general solution of the differential equation, we need to find the integrating factor of the differential equation and then use the standard method to solve the problem.

Complete step-by-step solution:
Given: The differential equation is given as:
dydx−y=sin⁡x.....(i)\dfrac{{dy}}{{dx}} - y = \sin x .....\left( i \right)
The given differential equation is of the form
dydx+p(x)y=Q(x).........(ii)\dfrac{{dy}}{{dx}} + p\left( x \right)y = Q\left( x \right) .........\left( {ii} \right)
On comparing (i) and (ii), we get

P(x)=−1and Q(x)=sin⁡x  P\left( x \right) = - 1 and \\\ Q\left( x \right) = \sin x \\\

The integrating factor is given by exp∫P(x)dx\int {P\left( x \right)} dx
∴\therefore integrating factor (I.F)=e∫−dx=e−x\left( {I.F} \right) = {e^{\int { - dx} }} = {e^{ - x}}
The general solution of the given form of differential equation is given by

y(I.F)=∫Q(x)  dx+C ⇒ye−x=∫sin⁡x dx+C ⇒ye−x=−cos⁡x+C  y\left( {I.F} \right) = \int {Q\left( x \right)} \,\,dx + C \\\ \Rightarrow y{e^{ - x}} = \int {\sin x} \,dx + C \\\ \Rightarrow {y^{e - x}} = - \cos x + C \\\

Hence, the general solution of the given differential equation is
ye−x=−cos⁡x+Cy{e^{ - x}} = - \cos x + C

Note: The students must keep in mind the standard forms of differential equation

dydx+P(x)y=Q(x), dxdy+P(y)x=Q(y)  \dfrac{{dy}}{{dx}} + P\left( x \right)y = Q\left( x \right), \\\ \dfrac{{dx}}{{dy}} + P\left( y \right)x = Q\left( y \right) \\\

Both these forms are extremely important. One must be very careful while finding the expressions for P(x),P(y),Q(x)  and  Q(y).P\left( x \right),P\left( y \right),Q\left( x \right)\,\,and\,\,Q\left( y \right). Choosing the wrong expression can lead to the calculation of wrong integrating factor and ultimately you will not be able to solve the problem. It is therefore advised to the students to solve as many problems of the two forms of differential equations. The Bernoulli equation is one of the well-known nonlinear differential equations of first order. In order to solve the Bernoulli differential equation, it is first converted into one of the two differential equation forms mentioned above by using a method of substitution. The new differential equation so obtained is solved by the same procedure used in our calculation and the final solution is obtained.