Pearson Edexcel IAL Mathematics FP2.6.4 Taylor series method for differential equations

Practise using differential equations and initial conditions to find derivative values, then express y as a Taylor series solution.

Exam points

differentiate a differential equation to express y''' or y'''' in terms of lower derivatives
substitute initial values into derivative formulae before forming a series solution
write the Taylor solution in powers of x or x-a up to the requested term

[Maximum number: 4]

Given that

y \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-4\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y=0

find a series solution for y in ascending powers of x, up to and including the term in $x^{3}$ , simplifying the coefficients where possible.

$\begin{aligned} \text { At } x=0 \quad \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}=\frac{4}{8}(1)^{2}-3=-\frac{5}{2} \quad \text { oe } \frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}=\frac{28}{64} \times 1^{3}-\frac{24}{8} \times 1=-\frac{41}{16} y=8+x-\frac{5}{2} \times \frac{x^{2}}{2!}-\frac{41}{16} \times \frac{x^{3}}{3!}+\ldots y=8+x-\frac{5}{4} x^{2}-\frac{41}{96} x^{3}+\ldots \end{aligned}$

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FP2.6.4 - Taylor series method for differential equations question 1