Standard +0.3 This is a separable first-order differential equation requiring partial fractions to decompose the right-hand side, followed by integration and applying initial conditions. While it involves multiple standard techniques (separation, partial fractions, integration, solving for y), each step is routine for P3 level with no novel insight required. Slightly above average difficulty due to the algebraic manipulation involved.
7 The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 1 } { ( x + 1 ) ( x + 3 ) }$$
It is given that \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
Separate variables correctly and integrate at least one side
B1
Obtain term \(\ln(y-1)\)
B1
Carry out a relevant method to determine \(A\) and \(B\) such that \(\dfrac{1}{(x+1)(x+3)} \equiv \dfrac{A}{x+1} + \dfrac{B}{x+3}\)
M1
Obtain \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\)
A1
Integrate and obtain terms \(\frac{1}{2}\ln(x+1) - \frac{1}{2}\ln(x+3)\)
A1 FT + A1 FT
FT is on \(A\) and \(B\)
Use \(x=0\), \(y=2\) to evaluate a constant, or as limits in a solution containing terms of the form \(a\ln(y-1)\), \(b\ln(x+1)\) and \(c\ln(x+3)\), where \(abc \neq 0\)
M1
Obtain correct answer in any form
A1
Obtain final answer \(y = 1 + \sqrt{\dfrac{3x+3}{x+3}}\), or equivalent
A1
## Question 7:
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables correctly and integrate at least one side | B1 | |
| Obtain term $\ln(y-1)$ | B1 | |
| Carry out a relevant method to determine $A$ and $B$ such that $\dfrac{1}{(x+1)(x+3)} \equiv \dfrac{A}{x+1} + \dfrac{B}{x+3}$ | M1 | |
| Obtain $A = \frac{1}{2}$ and $B = -\frac{1}{2}$ | A1 | |
| Integrate and obtain terms $\frac{1}{2}\ln(x+1) - \frac{1}{2}\ln(x+3)$ | A1 FT + A1 FT | FT is on $A$ and $B$ |
| Use $x=0$, $y=2$ to evaluate a constant, or as limits in a solution containing terms of the form $a\ln(y-1)$, $b\ln(x+1)$ and $c\ln(x+3)$, where $abc \neq 0$ | M1 | |
| Obtain correct answer in any form | A1 | |
| Obtain final answer $y = 1 + \sqrt{\dfrac{3x+3}{x+3}}$, or equivalent | A1 | |
---
7 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 1 } { ( x + 1 ) ( x + 3 ) }$$
It is given that $y = 2$ when $x = 0$.\\
Solve the differential equation, obtaining an expression for $y$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2020 Q7 [9]}}