Solving a simple differential equation.

Solve \[ y' = \frac{x}{1+x^2}, \quad y(0) = 1 \]

Integrate both sides

\[ \begin{align} y &= \int \frac{x}{1+x^2} dx \\ &= \frac{1}{2}\int \frac{2x}{1+x^2} dx \\ &= \frac{1}{2} \log(1+x^2) + c \end{align} \ \]

Use the initial condition to find the value of c.

\[ \begin{align} 1 &= \frac{1}{2} \log(1+0) + c \\ 1 &= c \\ \end{align} \ \] so c = 1 and \[ \begin{align} y &= \frac{1}{2} \log(1+x^2) + 1 \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

Solve \[ y' = e^{x/2}, \quad y(0) = 1 \]