| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Challenging +1.3 This is a Further Maths question requiring students to verify a given integrating factor (involving algebraic manipulation with surds) and then solve the differential equation. While the integrating factor form is non-standard and requires careful algebraic work with √(x²+1), the verification is guided and the solution method is systematic once the integrating factor is established. The 11 marks reflect substantial working, but this is a standard Further Maths technique without requiring novel insight. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks |
|---|---|
| 7(a) | dy y x ( ) |
| Answer | Marks |
|---|---|
| dx √(x2+1) x2+1 | B1 |
| Answer | Marks |
|---|---|
| e | M1 A1 |
| Answer | Marks |
|---|---|
| x 1) | A1 |
| Answer | Marks |
|---|---|
| 7(b) | d ( ( )) x |
| Answer | Marks |
|---|---|
| dx x2 +1 | M1 A1 |
| Answer | Marks |
|---|---|
| x2 +1 2 | M1 A1 |
| ln2=C | M1 |
| Answer | Marks |
|---|---|
| x+√(x2+1) 2 | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(a) ---
7(a) | dy y x ( )
+ = x−√(x2+1)
dx √(x2+1) x2+1 | B1
1
dx
√(x2+1) =esinh−1x
e | M1 A1
= +√( x2+
x 1) | A1
4
--- 7(b) ---
7(b) | d ( ( )) x
y x+√(x2 +1) =−
dx x2 +1 | M1 A1
( ) x ( )
y x+√(x2 +1) = − dx = − 1ln x2 +1 +C
x2 +1 2 | M1 A1
ln2=C | M1
( )
ln2−1ln x2+1
( ) ( )
y= 2 = x−√(x2+1) ln 1√(x2+1)
x+√(x2+1) 2 | M1 A1
7
Question | Answer | Marks\begin{enumerate}[label=(\alph*)]
\item Show that an appropriate integrating factor for
$$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$
is $x + \sqrt{x^2 + 1}$. [4]
\item Hence find the solution of the differential equation
$$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$
for which $y = \ln 2$ when $x = 0$. Give your answer in the form $y = f(x)$. [7]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q7 [11]}}