 Hi, and welcome to the session. I'm Shashi and I'm going to help you with the following question. Question is if y is equal to a multiplied by e raised to the power of mx plus b multiplied by e raised to the power nx, show that d square y upon dx square minus m plus n multiplied by dy by dx plus m and y is equal to 0. Let us start the solution now. We have given y is equal to a multiplied by e raised to the power mx plus b multiplied by e raised to the power nx. Now differentiating both sides with respect to x we get dy upon dx is equal to a multiplied by m multiplied by e raised to the power mx plus b multiplied by n multiplied by e raised to the power nx. Now again differentiating both sides with respect to x we get d square y upon dx square is equal to a m square e raised to the power mx plus b n square e raised to the power nx. Now let us find the value of d square y upon dx square minus m plus n dy upon dx plus mn y. Now substituting for d square y upon dx square dy upon dx and y we get this expression equal to d square y upon dx square is equal to this expression so we can write a m square e raised to the power mx plus b n square e raised to the power nx minus m plus n multiplied by dy by dx. Now dy by dx is equal to this expression so we can write a m multiplied by e raised to the power mx plus b n multiplied by e raised to the power nx plus mn multiplied by y we know y is equal to a e raised to the power mx plus b e raised to the power nx. Now on simplify we get a m square e raised to the power mx plus b n square e raised to the power nx minus a m square e raised to the power mx plus minus a mn e raised to the power mx minus b mn e raised to the power nx minus b n square e raised to the power nx plus a mn e raised to the power mx plus b mn e raised to the power nx. Now a m square e raised to the power mx minus a m square e raised to the power mx we get cancelled. Similarly, bn square e raised to the power nx minus bn square e raised to the power nx will cancel each other then minus bmn e raised to the power nx plus bmn e raised to the power nx will get cancelled then amn e raised to the power nx minus amn e raised to the power power mx will cancel each other. Now our final answer is equal to 0, so we can write therefore d square by upon dx square minus m plus n multiplied by dy upon dx plus mny is equal to 0. We were required to prove this only, so hence proved. This completes the session. Hope you understood the session. Take care and goodbye.