General Forms (Analytical Solution)
Select a common differential equation and enter the parameters to see its solution, graph, and steps.
\( \displaystyle \frac{dy}{dx} = ky \)
Exponential Type
Enter Parameters
\( y(x) = y_0 \cdot e^{kx} \)
Graph Plot
Enter graph range
Solution Steps
\[ \frac{dy}{dx} = ky \]
Separate the variables:
\[ \frac{1}{y} dy = k dx \]
Integrate both sides:
\[ \int \frac{1}{y} dy = \int k dx \]
\[ \ln |y| = kx + C \]
Exponentiate both sides:
\[ |y| = e^{kx + C} = e^{kx} \cdot e^C = C' e^{kx} \]
Using the initial condition \( y(0) = y_0 \):
\[ y_0 = C' e^0 = C' \]
Therefore, the solution is: