CAIE A-Level Further Mathematics 2.6.6 Initial Conditions

Practise substituting given function and derivative values into a complete differential-equation solution to determine constants and interpret its later behaviour.

Exam points

form the full general solution before substituting the given value of the function and its derivative
solve the resulting simultaneous equations for every arbitrary constant and state the particular solution
for large input values, remove decaying terms and rewrite the surviving oscillation in amplitude-phase form

[Maximum number: 7]

Hence find the solution of the differential equation

\left(x^{2}+1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+y \sqrt{x^{2}+1}=x^{2}-x \sqrt{x^{2}+1}

for which $y=\ln 2$ when x=0. Give your answer in the form y=f(x).

$\frac{\mathrm{d}}{\mathrm{d} x}\left(y\left(x+\sqrt{ }\left(x^{2}+1\right)\right)\right)=-\frac{x}{x^{2}+1}$

M1 A1

$y\left(x+\sqrt{ }\left(x^{2}+1\right)\right)=-\int \frac{x}{x^{2}+1} \mathrm{~d} x=-\frac{1}{2} \ln \left(x^{2}+1\right)+C$

M1 A1

$\ln 2=C$

M1

$y=\frac{\ln 2-\frac{1}{2} \ln \left(x^{2}+1\right)}{x+\sqrt{ }\left(x^{2}+1\right)}=\left(x-\sqrt{ }\left(x^{2}+1\right)\right) \ln \left(\frac{1}{2} \sqrt{ }\left(x^{2}+1\right)\right)$

M1 A1

7b 7

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2.6.6—Initial conditions question 1