 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, find the equation of a curve passing through the point 00 and whose differential equation is y dash is equal to e raised to the power x sin x. Let us now start with the solution. Given differential equation is y dash is equal to e raised to the power x multiplied by sin x. Now we know y dash represents dy upon dx. So we can replace y dash by dy upon dx in this equation and we get dy upon dx is equal to e raised to the power x multiplied by sin x. Now separating the variables in this equation we get dy is equal to e raised to the power x sin x dx. Now integrating both the sides of this equation we get integral of dy is equal to integral of e raised to the power x sin x dx. Now using this formula of integration we can find this integral and this integral is equal to y. So we can write y is equal to i plus c where i is equal to integral of e raised to the power x sin x dx and c represents the constant of integration. Let us name this equation as equation 1. Now we will evaluate this integral. Now we can find this integral by using integration by parts. Now we can write i is equal to this is first function and this is second function. So i is equal to first function multiplied by integral of second function minus integral of derivative of first function multiplied by integral of second function dx. Now using this formula of integration we get integral of sin x dx is equal to minus cos x. So we will replace this integral by minus cos x. Now we will write this minus sign as it is. Here we will write integral of derivative of e raised to the power x is e raised to the power x only and integral of sin x dx is minus cos x. So here we can write minus cos x and we will write this dx as it is. Now i is further equal to minus e raised to the power x cos x plus integral of e raised to the power x cos x dx. Now again we will find this integral by using integration by parts. We will write this term as it is. Here this is the first function and this is the second function. So we can write first function multiplied by integral of second function minus integral of derivative of first function multiplied by integral of second function dx. Now using this formula of integration we get integral of cos x dx is equal to sin x. So we can write minus e raised to the power x cos x plus e raised to the power x sin x minus integral of e raised to the power x sin x dx. Clearly we can see here we have written this term as it is. We have replaced integral of cos x dx by sin x. Derivative of e raised to the power x is e raised to the power x and again integral of cos x dx is sin x and we have written this dx as it is. Now we know integral of e raised to the power x sin x dx is equal to i. So we can substitute i for this integral. Now we get i is equal to minus e raised to the power x cos x plus e raised to the power x sin x minus i. Now adding i on both the sides of this equation we get 2i is equal to e raised to the power x multiplied by sin x minus cos x. Taking e raised to the power x common in these two terms we get this expression. Now dividing both the sides of this equation by 2 we get i is equal to e raised to the power x upon 2 multiplied by sin x minus cos x. Now we will substitute value of i in this equation. So we can write substituting i in equation 1 we get y is equal to e raised to the power x upon 2 multiplied by sin x minus cos x plus c. Now we know we are given that curve passes through origin that is it passes through 0 0. So to find the value of c we will substitute y and x equal to 0 in this equation. Now let us name this equation as equation 2. Now substituting x is equal to 0 and y is equal to 0 in equation 2 we get 0 is equal to e raised to the power 0 upon 2 multiplied by sin 0 minus cos 0 and here we will write this plus c as it is. Now this further implies 0 is equal to 1 upon 2 multiplied by 0 minus 1. Plus c we know e raised to the power 0 is equal to 1 sin 0 is equal to 0 and cos 0 is equal to 1 and we have written this minus sign as it is and plus c is also as it is. Now this further implies 0 is equal to minus 1 upon 2 plus c. Now adding 1 upon 2 on both the sides of this equation we get 1 upon 2 is equal to c. Or we can simply write c is equal to 1 upon 2. Now we will substitute this value of c in equation 2. Now we can write substituting c is equal to 1 upon 2 in equation 2 we get y is equal to 1 upon 2 multiplied by e raised to the power x multiplied by sin x minus cos x plus 1 upon 2. Now multiplying both the sides of this equation by 2 we get 2y is equal to e raised to the power x sin x minus cos x plus 1. Now subtracting 1 from both the sides of this equation we get 2y minus 1 is equal to e raised to the power x multiplied by sin x minus cos x. So this is the required equation of the curve. This completes the session. Hope you understood the solution. Take care and have a nice day.