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Differential Equations

Quick questions on Second-order linear differential equations explained: H2 Further Mathematics

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the auxiliary equation?
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Trying a solution of the form y=eλxy = \mathrm{e}^{\lambda x} and substituting gives, after dividing by eλx\mathrm{e}^{\lambda x}, the auxiliary equation
What are the three cases?
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In each case there are two arbitrary constants AA and BB.
What are applying initial conditions?
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A second-order equation needs two conditions, typically a value of yy and of dydx\dfrac{\mathrm{d}y}{\mathrm{d}x} at the same point. Differentiate the general solution, substitute both conditions, and solve the resulting simultaneous equations for AA and BB.
What are wrong solution form for complex roots?
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Complex roots α±βi\alpha \pm \beta i give eαx(Acos⁡βx+Bsin⁡βx)\mathrm{e}^{\alpha x}(A\cos\beta x + B\sin\beta x); do not leave the answer with e(α+βi)x\mathrm{e}^{(\alpha + \beta i)x}.
What is sign of the real part?
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In eαx\mathrm{e}^{\alpha x}, a negative α\alpha means decay; reading the sign wrongly inverts growth and decay.
What is q1?
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Write the auxiliary equation of d2ydx2−dydx−6y=0\dfrac{\mathrm{d}^2 y}{\mathrm{d}x^2} - \dfrac{\mathrm{d}y}{\mathrm{d}x} - 6y = 0. [1 mark]
What is q2?
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State the general solution form for complex auxiliary roots −1±2i-1 \pm 2i. [2 marks]
What is q3?
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What solution form corresponds to a repeated auxiliary root λ=4\lambda = 4? [1 mark]

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