Edexcel A-Level Mathematics A2 Fp2 5 2 Reducible Second Order Differential Equations Questions

Practise Edexcel IAL FP2.5.2 by using chain-rule substitutions for derivatives, obtaining constant-coefficient equations and converting solutions back to x.

Exam points

use the chain rule to express dy/dx and d²y/dx² in the new variable
show the transformed constant-coefficient equation before solving it
convert the transformed solution back to x and state the general solution

[Maximum number: 2]

(a)

Hence show that the transformation $t=\ln x$ , where x>0, transforms the differential equation

x^2\frac{d^2y}{dx^2}-2y=1+4\ln x-2(\ln x)^2

into the differential equation

\frac{d^2y}{dt^2}-\frac{dy}{dt}-2y=1+4t-2t^2

[ 1 ]

x^2\frac{d^2y}{dx^2}-2y=e^{2t}e^{-2t}\left(\frac{d^2y}{dt^2}-\frac{dy}{dt}\right)-2y

= =\frac{d^2y}{dt^2}-\frac{dy}{dt}-2y.

Also $\ln x=t$ , so

\frac{d^2y}{dt^2}-\frac{dy}{dt}-2y=1+4t-2t^2

(b)

Hence determine the general solution of differential equation (I).

[ 1 ]

Since $t=\ln x$ ,

e^{2t}=x^2,\qquad e^{-t}=x^{-1}

Therefore the general solution is

y=Ax^2+Bx^{-1}+(\ln x)^2-3\ln x+2

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Edexcel A-Level Mathematics A2 Fp2 5 2 Reducible Second Order Differential Equations Questions question 1

Question (a)

Question (b)