Moderate -0.8 This is a straightforward separable differential equation requiring simplification of the right-hand side to 1/(1+x²), integration to arctan(x)+C, and expressing y explicitly. The algebraic simplification is immediate and the integration is a standard result. Below average difficulty due to minimal steps and routine techniques.
Attempt to separate variables. Attempt to use partial fractions of the form
M1*, M1*
\(\frac{A}{x} + \frac{Bx + c}{1 + x^2}\)
\(A = 1\)
A1
\(B = -1\)
A1
\(C = 0\)
A1
Obtain \(\ln y\)
B1
Obtain \(\ln x - \frac{1}{2}\ln(1 + x^2)\)
B1
Attempt to combine logs
M1*
Attempt to find an eqn not including logs
Attempt to deal with \(+c\)
M1*
Must be valid use of log or its inverse
Obtain \(y = \frac{Cx}{\sqrt{1 + x^2}}\)
A1
[10]
Attempt to separate variables. Attempt to use partial fractions of the form | M1*, M1* |
$\frac{A}{x} + \frac{Bx + c}{1 + x^2}$ |
$A = 1$ | A1 |
$B = -1$ | A1 |
$C = 0$ | A1 |
Obtain $\ln y$ | B1 |
Obtain $\ln x - \frac{1}{2}\ln(1 + x^2)$ | B1 |
Attempt to combine logs | M1* | Attempt to find an eqn not including logs
Attempt to deal with $+c$ | M1* | Must be valid use of log or its inverse
Obtain $y = \frac{Cx}{\sqrt{1 + x^2}}$ | A1 | [10] |
Find the general solution of the differential equation
$$\frac{dy}{dx} = \frac{x}{x(1 + x^2)}$$
giving your answer in the form $y = f(x)$. [10]
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q12 [10]}}