Standard +0.3 This is a standard separable differential equation requiring partial fractions decomposition of 1/(x²(1+2x)), followed by integration and applying initial conditions. While it involves multiple techniques (separation, partial fractions, integration, rearrangement), these are all routine procedures for P3/Further Pure students with no novel insight required. The partial fractions setup is straightforward, making this slightly easier than average.
10 The variables \(x\) and \(t\) satisfy the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 + 2 x )\), and \(x = 1\) when \(t = 0\).
Using partial fractions, solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
State a suitable form of partial fractions for \(\frac{1}{x^2(1+2x)}\)
B1
e.g. \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{1+2x}\) or \(\frac{Ax+B}{x^2} + \frac{C}{1+2x}\)
Use a relevant method to determine a constant
M1
Obtain one of \(A = -2\), \(B = 1\) and \(C = 4\)
A1
Obtain a second value
A1
Obtain the third value
A1
Separate variables correctly and integrate at least one term
M1
Obtain terms \(-2\ln x - \frac{1}{x} + 2\ln(1+2x)\) and \(t\)
B3 FT
The FT is on \(A\), \(B\) and \(C\). Withhold B1 for each error or omission
Evaluate a constant, or use limits \(x=1\), \(t=0\) in a solution containing terms \(t\), \(a\ln x\) and \(b\ln(1+2x)\), where \(ab \neq 0\)
M1
Obtain a correct expression for \(t\) in any form, e.g. \(t = -\frac{1}{x} + 2\ln\left(\frac{1+2x}{3x}\right) + 1\)
A1
## Question 10:
| Answer | Mark | Guidance |
|--------|------|----------|
| State a suitable form of partial fractions for $\frac{1}{x^2(1+2x)}$ | B1 | e.g. $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{1+2x}$ or $\frac{Ax+B}{x^2} + \frac{C}{1+2x}$ |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of $A = -2$, $B = 1$ and $C = 4$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Separate variables correctly and integrate at least one term | M1 | |
| Obtain terms $-2\ln x - \frac{1}{x} + 2\ln(1+2x)$ and $t$ | B3 FT | The FT is on $A$, $B$ and $C$. Withhold B1 for each error or omission |
| Evaluate a constant, or use limits $x=1$, $t=0$ in a solution containing terms $t$, $a\ln x$ and $b\ln(1+2x)$, where $ab \neq 0$ | M1 | |
| Obtain a correct expression for $t$ in any form, e.g. $t = -\frac{1}{x} + 2\ln\left(\frac{1+2x}{3x}\right) + 1$ | A1 | |
10 The variables $x$ and $t$ satisfy the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 + 2 x )$, and $x = 1$ when $t = 0$.\\
Using partial fractions, solve the differential equation, obtaining an expression for $t$ in terms of $x$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2021 Q10 [11]}}