To verify y = g(x) is a solution to differential equation:

1. Find y', y'', etc.
2. Substitute into differential equation
3. Verify equality holds

Graphical representation showing $\frac{dy}{dx}$ at various points. Each small segment shows slope of solution at that point.

Solution curves follow the slope field pattern.

Euler's Method Formula and Iteration

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$

where h = step size, starting from $(x_0, y_0)$.

For $\frac{dy}{dx} = g(x)h(y)$:

1. Separate: $\frac{1}{h(y)}dy = g(x)dx$
2. Integrate both sides: $\int \frac{1}{h(y)}dy = \int g(x)dx$
3. Solve for y (if possible)

After finding general solution with +C, use initial condition y(x0) = y0 to find C.

Substitute initial values into general solution.

dy/dt = ky

Solution: $y = Ce^{kt}$

• Growth: k > 0 (population, compound interest)
• Decay: k < 0 (radioactive decay, cooling)

$\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$

where L = carrying capacity.

Solution: $P(t) = \frac{L}{1 + Ce^{-kt}}$

Maximum sustainable population L. Growth rate maximum when P = L/2.