Substitution z = x + y or similar linear combination

Questions where a substitution of the form z = ax + by or v = y - f(x) transforms the ODE into a separable or linear equation.

2 questions · Challenging +1.0

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CAIE Further Paper 2 2023 June Q2
7 marks Challenging +1.2
2 Use the substitution \(z = x + y\) to find the solution of the differential equation $$\frac { d y } { d x } = \frac { 1 + 3 x + 3 y } { 3 x + 3 y - 1 }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\operatorname { aln } ( \mathrm { x } + \mathrm { y } ) + \mathrm { b } ( \mathrm { x } - \mathrm { y } ) + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be determined.
OCR FP3 Q3
7 marks Standard +0.8
  1. Use the substitution \(z = x + y\) to show that the differential equation $$\frac{dy}{dx} = \frac{x + y + 3}{x + y - 1} \qquad (A)$$ may be written in the form \(\frac{dz}{dx} = \frac{2(z + 1)}{z - 1}\). [3]
  2. Hence find the general solution of the differential equation (A). [4]