 Hello and welcome to the session. In this session we discuss the following question which says, solve the following differential equation dy by dx plus y is equal to cos x minus sin x. Let's see the solution now. The given differential equation is dy by dx plus y is equal to cos x minus sin x. This given differential equation is a linear differential equation. Consider the linear differential equation of the form dy by dx plus py is equal to q. Now comparing these two differential equations we get p is equal to 1 and q is equal to cos x minus sin x. Now the integrating factor that is if is equal to e to the power integral p dx that is equal to e to the power integral 1 into dx. Or you can say this is equal to e to the power x that is we have the integrating factor if is equal to e to the power x. So we have the required solution of the differential equation y into the integrating factor is equal to integral q into integrating factor dx plus c. Now substituting the values for integrating factor q we get this is further equal to y into e to the power x is equal to integral q that is cos x minus sin x. This whole multiplied by e to the power x dx plus c where the c is the constant of integration. This further gives y into e to the power x is equal to integral cos x into e to the power x dx minus integral sin x into e to the power x dx plus c. So further we get y into e to the power x is equal to now let's solve this integral by parts. So we take this as the first function and this as the second function. So this is equal to first function that is cos x into integral of the second function that is e to the power x dx minus integral of differential of the first function that is cos x. Which would be minus sin x into integral of the second function that is e to the power x which is e to the power x into dx minus integral of sin x into e to the power x dx that is this term remains as it is plus c. So this further gives us y into e to the power x is equal to cos x into now integral of e to the power x dx is e to the power x plus integral of sin x into e to the power x dx minus integral of sin x into e to the power x dx plus c. Now this term cancels with this term and so we get y into e to the power x is equal to cos x into e to the power x plus c or you can say we get y is equal to cos x plus c into e to the power minus x. So this is the solution of the given differential equation. So this completes the session hope you have understood the solution of this question.