 Hi and how are you all today? The question says, solve the differential equation 1 plus e raised to the power 2x into dy plus e raised to the power x bracket 1 plus y square bracket close dx is equal to 0. We are given that y is equal to 1 when x is equal to 0. So let us proceed with solving this differential equation. Let us rewrite the given differential equation once again. We have 1 plus e raised to the power 2x in bracket multiplied by dy plus e raised to the power x bracket 1 plus y square bracket close dx is equal to 0. Now we have further making it into the form of dy by dx. So we have minus e raised to the power x 1 plus y square upon 1 plus e raised to the power 2x, right? Further, we can also write it as, I have minus e raised to the power x gets multiplied by dx, then we have dy upon 1 plus e raised to the power 2x and here this was also divided by 1 plus e raised to the power 2x over here. Further, if we put e raised to the power x equal to t, this implies we have e raised to the power t dx equal to exactly dt. So on substituting we have this will become minus dt upon 1 plus t square equal to dy upon 1 plus y square, right? Now here 1 upon 1 plus t square is equal to what? It is tan inverse t and we have a negative sign with it also. Similarly, here we have tan inverse y plus c. Now on substituting back the values we have minus tan inverse, the value of t is e raised to the power x equal to tan inverse y plus c. So on taking tan inverse e raised to the power x from left-hand side to right-hand side we have now tan inverse y plus tan inverse e raised to the power x equal to c. Now it is given to us in the question that when x is equal to 0 then y is equal to 1, right? So we have tan inverse 1 plus tan inverse e raised to the power 0 equal to c. So we have its value as pi by 4 plus pi by 4 equal to c which is pi by 2 is equal to c. So on substituting the value of c here we have now the answer to this question tan inverse y plus tan inverse e raised to the power x plus c which is pi by 2, right? So this is the answer to the given differential equation. This completes the session. Hope you understood it well and have a nice day.