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I have posted a link there to this topic so the OP can see my work.What is the particular solution to y'x-1=0 using the inition condition x=1,y=2?

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I have posted a link there to this topic so the OP can see my work.What is the particular solution to y'x-1=0 using the inition condition x=1,y=2?

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We are given the IVP:

\(\displaystyle y'x-1=0\) where \(\displaystyle y(1)=2\)

Separating the variables, we obtain:

\(\displaystyle dy=\frac{1}{x}\,dx\) where \(\displaystyle x\ne0\)

Switching the dummy variables of integration and utilizing the boundaries as limits, we obtain:

\(\displaystyle \int_2^{y(x)}\,du=\int_1^x\frac{dv}{v}\)

Applying the anti-derivative form of the FTOC, we find:

\(\displaystyle y(x)-2=\ln|x|\)

\(\displaystyle y(x)=\ln|x|+2\)