Advanced level 2025 North West Regional Mock mathematics mechanics 2 solutions
Advanced level 2025 North West Regional Mock mathematics mechanics 2 solutions
Solution: For the non-homogeneous term sin(x), we’ll use the method
of undetermined coefficients. Since sin(x) appears on the right-hand
side, we try a particular solution of the form:
| yp(x) = A sin(x) + B cos(x) Taking derivatives: yp′ (x) = A cos(x) |
(9) |
| (10) |
– B sin(x) yp′′(x) =
| –A sin(x) – B cos(x) Substituting into the original differential equation: –A sin(x) – B cos(x) + A cos(x) – B sin(x) = sin(x) (–A – B) sin(x) + (A – B) cos(x) = sin(x) Comparing coefficients: –A – B = 1 (coefficient of sin(x)) A – B = 0 (coefficient of cos(x)) |
(11) |
| (12) (13) |
|
| (14) (15) |
|
| From the second equation, we get A = B. Substituting into the first | |
| equation: –A – A = 1 –2A = 1 A = – |
(16) (17) (18) |
1
2
Therefore, A = B = –1 2, and our particular solution is:
yp(x) = –1
2
sin(x) – 1
2
cos(x) (19)
