How to solve $xy=2\int_1^xy(t)dt+5$?

Question

Could you please give me some hint how to solve this equation: $xy=2\int_1^xy(t)dt+5$.

It is not known whether $y(x)$ is continuous or not, so I could not use Fundamental Theorem of Calculus for differentiating both sides of this equation and thus transform integral equation onto differential one.

Thanks.

Answer 1

Derivate the equation and use fundamental theorem of calculus to obtain $y(x)+xy'(x)=2y(x)$. then $xy'(x)-y(x)=0$ devide it by $x^2$ , $\frac{y'(x)}{x^2}-\frac{y(x)}{x}=0$, hence $(\frac{y(x)}{x})'=0$ then $\frac{y(x)}{x}=c$, so $y(x)= cx$. substitute in the original equation you have $cx^2=2\int_{1}^{x}ctdt+5=cx^2-c+5$ and hence $c=5$. then $y(x)=5x$.