CAIE A-Level Further Math A2 2.6.4 Linear Differential Equations Questions

$6 m^{2}+5 m+1=0 \Rightarrow m=-\frac{1}{2},-\frac{1}{3}$
Auxiliary equation.

$x=A \mathrm{e}^{-\frac{1}{2} t}+B \mathrm{e}^{-\frac{1}{3} t}$
Complementary function. Allow with ' x= ' missing.

$x=p+q t+r t^{2} \Rightarrow x^{\prime}=q+2 r t \Rightarrow x^{\prime \prime}=2 r$
Particular integral and its derivatives.

$12 r+5 q+10 r t+p+q t+r t^{2}=t^{2}+10 t+13$
Substitutes and equates coefficients.

$r=1, \quad q=0, \quad p=1$$x=A \mathrm{e}^{-\frac{1}{2} t}+B \mathrm{e}^{-\frac{1}{3} t}+t^{2}+1$
Must have x=.

6

$x=t^{2}+1$

B1FT

Must have ' x= '. Accept ' x is approximately' but do
not accept ' $x \rightarrow{ }^{\prime}$. FT on their PI.

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