Pearson Edexcel IAL Mathematics FP2.5.2 Reducible second order differential equations

Practise reducing second order differential equations with a given substitution, then solving and converting back to the original variable.

Exam points

[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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