Boundary Value Problems/Introduction to BVPs/IVP-student-1

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Problem: Differential equation: ${\displaystyle y'=xy}$ with initial condition: ${\displaystyle y(0)=3}$

Start Solution:

${\displaystyle \int {\frac {1}{y}}dy=\int xdx}$

${\displaystyle ln(|y|)={\frac {x^{2}}{2}}+C}$

${\displaystyle e^{ln(|y|)}=e^{{\frac {x^{2}}{2}}+C}}$

${\displaystyle |y|=e^{\frac {x^{2}}{2}}e^{C}}$

${\displaystyle |y|=e^{\frac {x^{2}}{2}}C_{1}}$

${\displaystyle |y|=C_{1}e^{\frac {x^{2}}{2}}}$

Use the initial condition ${\displaystyle y(0)=3}$ to solve for the constant ${\displaystyle C_{1}}$.

${\displaystyle |y(0)|=3=C_{1}e^{0}}$

${\displaystyle C_{1}=3}$

The particular solution to the initial value problem is:

${\displaystyle y=3e^{\frac {x^{2}}{2}}}$ Why did I drop the "absolute value" operation?

End Solution:

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