Kansas State University

Mathematics Department

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First Order Linear Equations

Additional Examples

Solve the following initial value problem $$ \begin{align} \frac{dy}{dx} + 10 y &= -7 x - 9 \\ y(0) &= 7 \end{align} $$ This is a linear equation. First we find the general solution following the paradigm.

  1. Find the integrating factor $$ \mu(x) = \exp(\int 10 dx) = \exp(10x) $$
  2. Multiply through by the integrating factor $$ \exp(10x) \frac{dy}{dx} + 10\exp(10x)y = (-7 x - 9)\exp(10x) $$
  3. Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(10x)y) =(-7 x - 9)\exp(10x) $$
  4. Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right.$$ \exp(10x)y = (-(7/10)x - 83/100)\exp(10x) + C $$
  5. Divide through by $\mu(x)$ to solve for $ y.$ $$y = -(7/10)x - 83/100+ C\exp(-10x) $$
Now we plug in the initial values $ x = 0 $ and $ y = 7$ and solve for $ C = 783/100$, to obtain the solution to the initial value problem $$ y = -(7/10)x - 83/100 + (783/100)\exp(-10x) $$ You may reload this page to generate additional examples.

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