 Hello friends and how are you all today? My name is Priyanka and let us discuss this question. It says solve the following differential equation Now we are given d square y upon dx square is equal to e raised to the power x plus cos x given that dy by dx is equal to 1 is equal to y when x is equal to 0. So here let us rewrite the given differential equation on integrating this equation with respect to x we get integral d square y upon dx square dx is equal to integral e raised to the power x plus cos x dx or we can write it as dy by dx is equal to Integral e raised to the power x dx plus integral cos x we have dy by dx is equal to e raised to the power x plus equation. Now we know that when x is equal to 0 dy by dx is equal to 1 is equal to e raised to the power 0 plus c1 this implies that the value of c1 is equal to. So now on substituting this value of c1 equal to 0 in the first equation we get dy by dx is equal to e raised to the power x plus sin x plus 0 or we can write it as dy by dx is equal to e raised to the power x plus sin x. Again integrating with respect to x we get dy by dx into dx is equal to integral e x dx plus integral sin x that is y is equal to e raised to the power x minus cos x plus c2. Let this be the second equation. Now we know that when x is equal to 0 y is equal to 1 so on substituting these values we have 1 is equal to e raised to the power 0 minus cos 0 plus c2. That implies 1 is equal to 1 minus 1 plus c2 that implies the value of c2 is equal to 1. So on substituting the value of c2 in second equation we get y is equal to e raised to the power x minus cos x plus 1. Right? So this is the required answer to the given question. So hope you understood it well and enjoyed it too. Have a nice day.