# reid's question at Yahoo! Answers regarding a first order separable IVP

#### MarkFL

Staff member
Here is the question:

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

#### MarkFL

Staff member
Hello reid,

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:

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

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