 Hi and welcome to the session. Let us discuss the following question, questions ways. Solve the differential equation. Given differential equation is y multiplied by e raised to the power x upon y dx is equal to x multiplied by e raised to the power x upon y plus y square dy where y is not equal to 0. Let us now start with the solution. Now the given differential equation is y multiplied by e raised to the power x upon y dx is equal to x multiplied by e raised to the power x upon y plus y square dy. Now dividing both the sides of this equation by this term that is y multiplied by e raised to the power x upon y we get dx is equal to x multiplied by e raised to the power x upon y plus y square dy upon y multiplied by e raised to the power x upon y. Now dividing both the sides of this equation by dy we get dx upon dy is equal to x multiplied by e raised to the power x upon y upon y multiplied by e raised to the power x upon y plus y square upon y multiplied by e raised to the power x upon y. Now this implies dx upon dy is equal to x upon y plus y upon e raised to the power x upon y. Here e raised to the power x upon y will cancel e raised to the power x upon y and here this y will cancel one y here and we get x upon y plus y upon e raised to the power x upon y is equal to dx upon dy. Now clearly we can see this is a homogeneous differential equation and degree of this homogeneous function is 0. So we will put v is equal to x upon y. Now this implies vy is equal to x. Now differentiating both the sides with respect to y we get v plus y multiplied by dv upon dy is equal to dx upon dy or we can simply write dx upon dy is equal to v plus y multiplied by dv upon dy. Now let us name this equation as equation 1. Now we will substitute this value of dx upon dy in equation 1. Let us name this expression as 2. Now substituting value of dx upon dy in equation 1 we get v plus y multiplied by dv upon dy is equal to v plus y upon e raised to the power v. We know we have substituted v for x upon y. Now we will cancel v from both the sides of this equation and we get y multiplied by dv upon dy is equal to y upon e raised to the power v. Or we can say subtracting v from both the sides of this equation we get this equation. Now dividing both the sides of this equation by y we get dv upon dy is equal to 1 upon e raised to the power v. Now separating the variables in this equation we get e raised to the power v dv is equal to dy. Now integrating both the sides of this equation we get integral of e raised to the power v dv is equal to integral of dy. Now we can find this integral by using this formula. So here we can write this integral is equal to e raised to the power v. Now we will write this is equal to sin as it is and we can find this integral by using this formula. Now we can write this integral is equal to y and here we can write plus c where c represents the constant of integration. Now replacing v by x upon y in this equation we get e raised to the power x upon y is equal to y plus c. So this is the required solution for the given differential equation. This completes the session. Hope you understood the solution. Take care and keep smiling.