Kansas State University

Mathematics Department

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First Order Linear Equations

Additional Examples

Solve the following initial value problem $$ \begin{align} \frac{dy}{dx} - 8 y &= -10 x - 2 \\ y(0) &= 0 \end{align} $$ This is a linear equation. First we find the general solution following the paradigm.

  1. Find the integrating factor $$ \mu(x) = \exp(\int -8 dx) = \exp(-8x) $$
  2. Multiply through by the integrating factor $$ \exp(-8x) \frac{dy}{dx} - 8\exp(-8x)y = (-10 x - 2)\exp(-8x) $$
  3. Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(-8x)y) =(-10 x - 2)\exp(-8x) $$
  4. Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right.$$ \exp(-8x)y = ((5/4)x + 13/32)\exp(-8x) + C $$
  5. Divide through by $\mu(x)$ to solve for $ y.$ $$y = (5/4)x + 13/32+ C\exp(8x) $$
Now we plug in the initial values $ x = 0 $ and $ y = 0$ and solve for $ C = -13/32$, to obtain the solution to the initial value problem $$ y = (5/4)x + 13/32 - (13/32)\exp(8x) $$ You may reload this page to generate additional examples.

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