Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (using substitution for the square root term), apply initial conditions, and rearrange for y. While it involves 6 marks and requires careful algebraic manipulation, it follows a completely standard template with no conceptual challenges or novel insights required—making it slightly easier than the average A-level question.
2. Find the particular solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y ^ { 2 } } { \sqrt { 4 x + 5 } } \quad x > - \frac { 5 } { 4 }$$
for which \(y = \frac { 1 } { 3 }\) at \(x = - \frac { 1 } { 4 }\) giving your answer in the form \(y = \mathrm { f } ( x )\)
(6)
Separates variables; "4" may be on either side but must be correct; \(dy\) and \(dx\) must be present and in correct place
\(-\frac{1}{y} = 2\sqrt{4x+5}\ (+c)\)
M1, A1
M1: integrates one side to correct form; look for \(\frac{a}{y}\) or \(k\sqrt{4x+5}\). A1: correct integration both sides, allow unsimplified, no requirement for \(+c\)
Substitutes \(y=\frac{1}{3},\ x=-\frac{1}{4} \Rightarrow -3 = 4 + c \Rightarrow c = \ldots\)
dM1
Dependent on having integrated one side correctly
Rearranges \(\frac{a}{y} = b\sqrt{4x+5} + c\) to \(y = \ldots\)
ddM1
Dependent on: integrating both sides, substituting to find \(c\), and rearranging using correct method; do not allow each term to be inverted
\(y = \dfrac{1}{7 - 2\sqrt{4x+5}}\)
A1
Or exact equivalent e.g. \(y = \dfrac{-1}{2(4x+5)^{0.5}-7}\); do not isw
# Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = \frac{4y^2}{\sqrt{4x+5}} \Rightarrow \int \frac{1}{y^2}\,dy = \int \frac{4}{\sqrt{4x+5}}\,dx$ | B1 | Separates variables; "4" may be on either side but must be correct; $dy$ and $dx$ must be present and in correct place |
| $-\frac{1}{y} = 2\sqrt{4x+5}\ (+c)$ | M1, A1 | M1: integrates one side to correct form; look for $\frac{a}{y}$ or $k\sqrt{4x+5}$. A1: correct integration both sides, allow unsimplified, no requirement for $+c$ |
| Substitutes $y=\frac{1}{3},\ x=-\frac{1}{4} \Rightarrow -3 = 4 + c \Rightarrow c = \ldots$ | dM1 | Dependent on having integrated one side correctly |
| Rearranges $\frac{a}{y} = b\sqrt{4x+5} + c$ to $y = \ldots$ | ddM1 | Dependent on: integrating both sides, substituting to find $c$, and rearranging using correct method; do not allow each term to be inverted |
| $y = \dfrac{1}{7 - 2\sqrt{4x+5}}$ | A1 | Or exact equivalent e.g. $y = \dfrac{-1}{2(4x+5)^{0.5}-7}$; do not isw |
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