Standard +0.8 This is a first-order linear differential equation requiring the integrating factor method with a non-trivial coefficient (4-x²), followed by integration involving partial fractions or substitution, and applying an initial condition. While the method is standard for Further Maths, the algebraic manipulation and integration steps are more demanding than typical A-level questions, placing it moderately above average difficulty.
12 Solve the differential equation \(\left( 4 - x ^ { 2 } \right) \frac { d y } { d x } - x y = 1\), given that \(y = 1\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).
Integral must come from an attempt to get on its own
𝑑𝑥
For integrating
Multiplying both sides by their IF
Rearranging into the form y=, equation must come from an
attempt at integration having used IF and include c
Substituting in x = 0, y = 1 to lead to a value of c
Question 12:
12 | d𝑦 𝑥 1
− 𝑦 =
d𝑥 4−𝑥2 4−𝑥2
− x d x
IF e 4 − x 2
1 ln(4−𝑥2)
= e 2
= √4−𝑥2
d 1
(√4−𝑥2 𝑦) =
d𝑥 √4−𝑥2
1
4 − x 2 y = d x
4 − x 2
𝑥
= arcsin +𝑐
2
1
arcsin( 𝑥)+𝑐
2
𝑦 =
√4−𝑥2
when 𝑥 = 0, 𝑦 = 1 ⇒ 𝑐 = 2
1
arcsin( 𝑥)+2
2
𝑦 =
√4−𝑥2 | B1
M1
M1
A1
M1
A1
M1
M1
A1
[9] | 2.1
2.1
2.1
2.2a
2.1
1.1
2.2a
2.1
2.2a | 𝑑𝑦
Integral must come from an attempt to get on its own
𝑑𝑥
For integrating
Multiplying both sides by their IF
Rearranging into the form y=, equation must come from an
attempt at integration having used IF and include c
Substituting in x = 0, y = 1 to lead to a value of c
12 Solve the differential equation $\left( 4 - x ^ { 2 } \right) \frac { d y } { d x } - x y = 1$, given that $y = 1$ when $x = 0$, giving your answer in the form $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q12 [9]}}