 Hello and welcome to the session, I am Deepika here. Let's discuss the question which says find the general solution of the following differential equation dy by dx plus 3y is equal to e raised to power minus 2x. Now the given differential equation is a first order linear differential equation because it is of the form dy by dx plus p y is equal to q where p and q are constants or functions of x only. Now this type of differential equations are solved when they are multiplied by a factor which is known as the integrating factor because by multiplication of this factor the left hand side of the differential equation becomes exact differential of some function and it is given by a raised to power integral of p dx and the solution of the given differential equation is given by y into integrating factor is equal to integral of q into integrating factor dx plus c. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now the given differential equation is dy by dx plus 3y is equal to e raised to power minus 2x. This is a linear differential equation dy by dx plus p y is equal to q where p is equal to 3 and q is equal to e raised to power minus 2x. Therefore according to our key idea integrating factor is equal to e raised to power integral of p dx and this is equal to e raised to power integral of 3 dx and this is equal to e raised to power 3x. On multiplying both sides of the given differential equation by the integrating factor which is equal to e raised to power 3x we get dy by dx into e raised to power 3x plus e raised to power 3x into 3y is equal to e raised to power minus 2x into e raised to power 3x or now the left hand side is the differential of y into e raised to power 3x. So dy dx of y into e raised to power 3x is equal to e raised to power x. Now integrating both sides with respect to x we have integral of d over dx of y into e raised to power 3x dx is equal to integral of e raised to power x dx or y into e raised to power 3x is equal to e raised to power x plus c or y is equal to e raised to power x over e raised to power 3x that is e raised to power minus 2x plus c into e raised to power minus 3x. Hence the general solution of the given differential equation is y is equal to e raised to power minus 2x plus c into e raised to power minus 3x. So this is our answer for the above question. This completes our session. I hope the solution is clear to you and you have enjoyed the session. Bye and take care.