 Good morning friends, I am Kudva and today we will discuss the following question find the general solution of the differential equation x plus 3 y square into dy by dx is equal to y and here y is greater than 0 Let us now begin with the solution. So the given differential equation is x plus 3 y square into dy by dx is Equal to y or we can write this as dx upon dy is Equal to x plus 3 y square upon y and This implies dx upon dy is Equal to x upon y plus 3 y and This further implies dx upon dy minus x upon y is Equal to 3 y. Let us mark this as equation 1 does Equation 1 is a linear differential equation of the form dx upon dy Plus p1 x is equal to q1 and here p1 and q1 Are functions of y? That's the integrating factor of Equation 1 is given by e raised to the power integral p1 dy And this is equal to e raised to the power integral Now comparing this equation dx upon dy plus p1 x is equal to q1 with this equation 1 That is dx upon dy minus x upon y is equal to 3 y We can clearly see that p1 is equal to minus 1 upon y So we have e to the power integral minus 1 upon y dy And this is equal to e raised to the power now e raised to the power integral minus 1 upon y gives e raised to the power minus log y And this is further equal to e raised to the power log 1 upon y And this is equal to 1 upon y because e raised to the power log 1 upon y can be written as 1 upon y Now multiplying each term of equation 1 by 1 upon y we get 1 upon y into dx upon dy Minus x upon y into 1 upon y is equal to 3 y into 1 upon y Now we cancel y in numerator and denominator here and we get this implies D upon dy of x into 1 upon y is equal to 3 Now on differentiating x into 1 upon y with respect to y we get 1 upon y into dx upon dy minus x upon y into 1 upon y So we can write this term as D upon dy of x into 1 upon y Now on integrating both the sides of the above equations with respect to y we get Integral D upon dy of x upon y dy is equal to integral 3 dy and this implies now on integrating D upon dy of x upon y we get x upon y and This is equal to now integrating 3 with respect to y we get 3 y plus C where C is a constant And this implies x is equal to 3 y square plus C y Hence the required solution is x is equal to 3 y square plus C y This is our answer. Hope you have understood the solution. Bye and take care