Standard +0.8 This is a first-order linear ODE requiring the integrating factor method, a standard FP3 technique. However, it involves non-trivial integration (requiring substitution u = 1 + x³) and careful algebraic manipulation to reach the final form. The 8-mark allocation and Further Pure content place it moderately above average difficulty, but it follows a well-defined algorithmic approach without requiring novel insight.
Find the solution of the differential equation
$$\frac{dy}{dx} - \frac{x^2y}{1 + x^3} = x^2$$
for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = f(x)\). [8]
For correct process for finding integrating factor; For correct IF, simplified (here or later); For multiplying through by their IF; For integrating RHS to obtain \(A(1+x^3)^k\) OR \(\ln A(1+x^3)^k\); For substituting (0, 1) into GS (including \(+c\)); For correct \(c\). f.t. from their GS; For correct solution. AEF in form \(y = f(x)\)
Integrating factor $e^{\int\frac{x^2}{1+x^3}dx}$; $= e^{\frac{1}{3}\ln(1+x^3)} = (1+x^3)^{-1}$; $\frac{d}{dx}\left[y(1+x^3)^{-1}\right] = \frac{x^2}{(1+x^3)^1}$; $y(1+x^3)^{-1} = \frac{1}{3}(1+x^3)^{\frac{2}{3}}(+c)$; $1 = \frac{1}{3}+c \Rightarrow c = \frac{1}{3}$; $y = \frac{1}{2}(1+x^3) + \frac{1}{2}(1+x^3)^{\frac{1}{3}}$ | M1; A1; M1; M1; M1 A1√; A1 [8] | For correct process for finding integrating factor; For correct IF, simplified (here or later); For multiplying through by their IF; For integrating RHS to obtain $A(1+x^3)^k$ **OR** $\ln A(1+x^3)^k$; For substituting (0, 1) into GS (including $+c$); For correct $c$. f.t. from their GS; For correct solution. **AEF** in form $y = f(x)$
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Find the solution of the differential equation
$$\frac{dy}{dx} - \frac{x^2y}{1 + x^3} = x^2$$
for which $y = 1$ when $x = 0$, expressing your answer in the form $y = f(x)$. [8]
\hfill \mbox{\textit{OCR FP3 2006 Q4 [8]}}