Solving Differential Equation: xy' + y = sqrt(x-1)

AI Thread Summary
The discussion centers on solving the differential equation xy' + y = sqrt(x-1), which the user struggles with due to their limited experience with only separable equations. They express difficulty in transforming the equation into a suitable form for integration, as it contains both y and x on the right side. A suggestion is made to use integrating factors, which are necessary for solving this type of differential equation. The user is encouraged to explore this method to find a solution. Understanding integrating factors is crucial for progressing in differential equations.
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i just started a course in differential equations, and this question was in the homework that i got, I am not sure we learned how to solve it but maybe i am meant to know.

xy' + y = sqrt(x-1)

up until now all i have been able to solve is equations which i can bring to a form of dx/x +dy/y = 0 or something like that

because this question has 3 parts i haven't been able to get F(x)dx + F(y)dy=0

xy' + y = sqrt(x-1)

x*dy/dx + y =sqrt(x-1)

dy/y + x*dx = sqrt(x-1)*dx/y

dy/y = (sqrt(x-1)/y - x )dx

now i can't integrat this because on the right i have y as well as x and cannot get rid of it
 
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Looks like you are familiar only with separable differential equations. Have you learned about integrating factors? That's what you need to solve this DE.
 

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