Second-order, constant coefficients

ก้าวต่อก้าว

1. y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

   The differential equation.

2. \text{order } 2

   Order 2: the highest derivative present.

3. \text{Constant coefficients, homogeneous}

   Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.

4. r^{2} + 1 = 0

   Characteristic equation of the homogeneous part (substitute y = e^{rx}).

5. r = - i,\ r = i

   Its roots.

6. y{\left(x \right)} = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}

   General solution (C₁, C₂ … are arbitrary constants).

7. \checkmark

   Verified: substituting the solution back into the equation gives 0.