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

   The differential equation.

2. \text{order } 2

   Order 2: the highest derivative present.

3. \text{Undetermined coefficients}

   Solve the homogeneous part from the characteristic equation, then guess a particular solution shaped like the right-hand side.

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)} - \frac{\sin{\left(2 x \right)}}{3}

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

7. \checkmark

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