Find the general solution of the differential equation d²x/dt² + 7dx/dt + 10x = 116 sin 2t. State an approximate solution for large positive values of t.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

1. [M1] Auxiliary equation: m² +7m+10=0⇒(m+2)(m+5)=0⇒ m = -2 or -5
2. [A1] CF: A℮⁻²ᵗ + B℮⁻⁵ᵗ
3. [M1A1] PI: Let x = p sin 2t + q cos 2t. ẋ = 2p cos 2t – 2q sin 2t. ẍ = −4p sin 2t – 4q cos 2t
4. [M1] Substitute into differential equation: (10p-4p-14q)sin 2t + (10q-4q+14p)cos 2t = 116 sin 2t. Equating coefficients: 6p - 14q = 116, 14p + 6q = 0
5. [M1A1] Solving the simultaneous equations: p = 3, q = −7 ⇒ x = 3 sin 2t - 7 cos 2t
6. [A1] GS: x = A℮⁻²ᵗ + B℮⁻⁵ᵗ + 3 sin 2t – 7 cos 2t
7. [B1] For large positive values of t, the exponential terms tend to 0, so x ≈ 3 sin 2t - 7 cos 2t.