Standard +0.3 This is a standard second-order linear differential equation with constant coefficients and exponential RHS. While it's a Further Maths topic (making it inherently harder than Core A-level), the solution method is completely routine: find the complementary function by solving the auxiliary equation, then find a particular integral. The auxiliary equation factorizes easily (m+3)(m-1)=0, and the particular integral requires only the standard trial solution Ae^x with adjustment since e^x appears in the CF. This is a textbook exercise requiring no novel insight, but rates slightly above average due to being Further Maths content.
Question 13:
13 | d 2 y2 d y
+ 2 − 3 y = 2 e x
d x d x
AE: 2 2 3 0 + − = 3 , 1 = −
CF: y = A e − 3 x + B e x
PI: y = C x e x
d y
= C ( e x + x e x )
d x
d 2 y2
= C ( 2 e x + x e x )
d x
C(2ex +xex)+2C(ex+xex)−3Cxex =2ex
4C =2C = 1
2
GS: y= Ae −3x+Bex+1xex
2 | M1
A1
M1
A1
A1
M1
A1
[7] | 2.1
2.1
2.1
1.1
1.1
2.1
2.2a
13 Find the general solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = 2 e ^ { x }$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q13 [7]}}