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First order linear equations

First order linear equations

IB Year 6 | Grade 12

Solve the following differential equations by using an Integrating Factor

  1. \(x \frac{\mathrm{d}y}{\mathrm{d}x} + y = x^2 + 1\)
  2. \(\frac{\mathrm{d}y}{\mathrm{d}x} - \frac{y}{x} = -x e^{-x}\)
  3. \(x^2 \frac{\mathrm{d}y}{\mathrm{d}x} + 2xy = \cos x\)
  4. \(x \frac{\mathrm{d}y}{\mathrm{d}x} + 2y = \frac{\sin x}{x}\)
  5. \(x \frac{\mathrm{d}y}{\mathrm{d}x} = y - x^2 e^{-x}\)

Solution:

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Answer:(a) \(y=\dfrac{x^2}{3}+1+\dfrac Cx\). (b) \(y=x(e^{-x}+C)\). (c) \(y=\dfrac{\sin x+C}{x^2}\). (d) \(y=\dfrac{C-\cos x}{x^2}\). (e) \(y=x(e^{-x}+C)\). Each formula first holds on an interval with \(x\ne0\). In (a), only \(C=0\) extends through zero; in (e), every \(C\) does. The original equations in (b)–(d) exclude zero. Constants are independent between parts.

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