 Hello and welcome to the session. I am Deepika and I am going to help you to solve the following question. The question says, solve the following differential equation. d-vibre dx plus 2y is equal to 6 into e raised to the power x. Now we know that the linear differential equation is of the form d-vibre dx plus p-y is equal to q, where p and q are constants or functions of x only to solve this differential equation. We will first find out the integrating factor which is given by e raised to the 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 d-vibre dx 2y is equal to 6 into e raised to the power x. So according to our key idea, the given differential equation is of the form d-vibre dx plus p-y is equal to q, where p is equal to 2 and q is equal to 6 into e raised to the power x. Therefore according to our key idea, integrating factor is equal to e raised to the power integral of p dx. Now p is equal to 2 so integrating factor is equal to e raised to the power integral of 2 dx, which is again equal to e raised to the power 2x. So again according to our key idea, solution of the given equation is given by y into integrating factor which is e raised to the power 2x is equal to integral of 2 which is 6 into e raised to the power x into integrating factor which is e raised to the power 2x dx. So y into e raised to the power 2x is equal to integral of 6 into e raised to the power 3x dx and this is equal to 6 into e raised to the power 3x over 3 plus c. Now this can be written as y into e raised to the power 2x is equal to 2 into e raised to the power 3x plus c or again it can be written as y is equal to 2 e raised to the power x plus c into e raised to the power minus 2x. So y is equal to 2 into e raised to the power x plus c into e raised to the power minus 2x is the general solution of the given differential equation. y is equal to 2 into e raised to the power x plus c into e raised to the power minus 2x is the answer for the above question. This completes our session. I hope the solution is clear to you. Bye and have a nice day.