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Differential equations
Differential equations are mathematical rules that describe how a quantity changes. Solving them means working backwards from the rate of change to find the original function that follows these rules.
Need to know
What you need to know
- **Case 1: Two distinct real roots, $m_1, m_2$** The solution is $y = Ae^{m_1x} + Be^{m_2x}$.
- **Case 2: One repeated real root, $m$** The solution is $y = (Ax+B)e^{mx}$.
- **Case 3: Complex conjugate roots, $p \pm iq$** The solution is $y = e^{px}(A\cos(qx) + B\sin(qx))$.
Explanation
Decoding the Rules of Change
- Find the Complementary Function (CF) by solving the associated homogeneous equation (setting the right-hand side to zero).
- Choose the correct form for the Particular Integral (PI) based on the function on the right-hand side of the original equation.
- Substitute the PI into the full differential equation to find the values of its unknown coefficients.
- Combine the CF and PI to get the General Solution, then use any given initial conditions to find the specific constants.