Edexcel A-Level Mathematics A2 Fp2 4 3 Reducible First Order Differential Equations Questions

Practise Edexcel IAL FP2.4.3 by applying substitutions, solving the resulting linear differential equation with an integrating factor and recovering the requested general solution.

Exam points

show the transformed linear equation before solving it by an integrating factor
convert back to y and give the general solution in the requested form

[Maximum number: 3]

Show that the substitution $y^{2}=\frac{1}{t}$ transforms the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+y=x y^{3}

into the differential equation

\frac{\mathrm{d} t}{\mathrm{~d} x}-2 t=-2 x

Since $y^2=\frac1t$ , $y=t^{-1/2}$ , so

\frac{dy}{dx}=-\frac12t^{-3/2}\frac{dt}{dx}

Substituting into $\frac{dy}{dx}+y=xy^3$ :

-\frac12t^{-3/2}\frac{dt}{dx}+t^{-1/2}=xt^{-3/2}

Multiplying by $-2t^{3/2}$ gives

\frac{dt}{dx}-2t=-2x

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Edexcel A-Level Mathematics A2 Fp2 4 3 Reducible First Order Differential Equations Questions question 1