Edexcel IAL Mathematics FP2.4.2 first order linear differential equations

Practise solving first order linear differential equations by rearranging into dy/dx + Py = Q and using an integrating factor.

Exam points

rearrange the equation into linear form and calculate the integrating factor e^∫Pdx
integrate the product equation and give y=f(x), z(x) or y^2 in the requested form

[Maximum number: 6]

Solve differential equation (II) and determine $y^{2}$ in terms of x.

The integrating factor is

I=e^{-\int 2\,dx}=e^{-2x}

te^{-2x}=-2\int xe^{-2x}\,dx

= =xe^{-2x}-\int e^{-2x}\,dx

= =xe^{-2x}+\frac12e^{-2x}+c.

Hence

t=x+\frac12+ce^{2x}

Since $t=\frac1{y^2}$ ,

y^2=\frac{1}{x+\frac12+ce^{2x}}

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