A curve is such that the gradient at a general point with coordinates $$\((x, y)\)$$ is proportional to $$\(\frac{y}{\sqrt{x+1}}\)$$. The curve passes through the points with coordinates $$\((0,1)\)$$ and $$\((3, \mathrm{e})\)$$. By setting up and solving a differential equation, find the equation of the curve, expressing $$\(y\)$$ in terms of $$\(x\)$$. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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Exam No：9709_s21_qp_32 Year：2021 Question No：7
Answer:
State equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=k \frac{y}{\sqrt{x+1}}\)
Separate variables correctly for their differential equation and integrate at least one side
Obtain \(\ln y\)
Obtain \(2[k] \sqrt{x+1}\)
Use \((0,1)\) and \((3, \mathrm{e})\) in an expression containing \(\ln y\) and \(\sqrt{x+1}\)
and a constant of integration to determine \(k\) and/or a constant of integration \(c\) (or use \((0,1),(3, \mathrm{e})\) and \((x, y)\) as limits on definite integrals)
Obtain \(k=\frac{1}{2}\) and \(c=-1\)
Obtain \(y=\exp (\sqrt{x+1}-1)\)
Separate variables correctly for their differential equation and integrate at least one side
Obtain \(\ln y\)
Obtain \(2[k] \sqrt{x+1}\)
Use \((0,1)\) and \((3, \mathrm{e})\) in an expression containing \(\ln y\) and \(\sqrt{x+1}\)
and a constant of integration to determine \(k\) and/or a constant of integration \(c\) (or use \((0,1),(3, \mathrm{e})\) and \((x, y)\) as limits on definite integrals)
Obtain \(k=\frac{1}{2}\) and \(c=-1\)
Obtain \(y=\exp (\sqrt{x+1}-1)\)
Knowledge points:
3.8.1 formulate a simple statement involving a rate of change as a differential equation (The introduction and evaluation of a constant of proportionality, where necessary, is included.)
3.8.2 find by integration a general form of solution for a first order differential equation in which the variables are separable (Including any of the integration techniques from topic 3.5 above.)
3.8.3 use an initial condition to find a particular solution
Solution:
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