# Solving a differential equation based on integrals

by dalta   Last Updated July 12, 2019 10:20 AM - source

It is my first week dealing with Differential Equations, and I am totally lost at solving the following equation:

$$\int^x_0(x-t)y(t)dt=2x+\int^x_0y(t)dt$$

Any help would be greatly appreciated!

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From $$\int^x_0(x-t)y(t)dt=2x+\int^x_0y(t)dt$$ we derive

$$x\int^x_0y(t)dt -\int_0^xty(t)dt=2x+\int^x_0y(t)dt.$$

If we differentiate we get

$$\int^x_0y(t)dt+xy(x)-xy(x)=2+y(x).$$

Hence

$$\int^x_0y(t)dt=2+y(x).$$

Differentiation once again yields

$$y(x)=y'(x).$$

Can you proceed ?

Fred
July 12, 2019 10:09 AM