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Mathematics LibreTexts

2.4: Separable Differential Equations

A differential equation is called separable if it can be written as

\[ f(y)\,dy = g(x)\,dx. \]

Steps To Solve a Separable Differential Equation

To solve a separable differential equation

  1. Get all the \(y\)'s on the left hand side of the equation and all of the \(x\)'s on the right hand side.
  2. Integrate both sides.
  3. Plug in the boundary conditions (e.g. given initial values) to find the constant of integration (\(C\)).
  4. Solve for \(y\).

Example \(\PageIndex{1}\)

Solve \( \dfrac{dy}{dx} = y(3 - x) \) with \( y(0 )= 5 \).

Solution

\[\begin{align} \dfrac{dy}{y} &= (3 - x) dx \\   \int \dfrac{dy}{y} &= \int (3-x) dx  \\   \ln\; y &= 3x - \dfrac{x^2}{2} + C \\   \ln 5 &= 0 + 0 + C   \\ C &= \ln\; 5 \\   y &= e^{3x-\frac{x^2}{2}} + \ln \; 5 \\ y &= e^{3x-\frac{x^2}{2}} \; e^{\ln \; 5} \\ &= 5e^{3x-\frac{x^2}{2}}.   \end{align}\]

Exercise  \(\PageIndex{1}\)

  1. \( \dfrac{dy}{dx} = \dfrac{x}{y} \) with \( y(0) = 1 \)
  2. \( \dfrac{dy}{dx} = x(x+1) \) with \( y(1) = 1  \)
  3. \( 2xy + \dfrac{dy}{dx} = x \)  with \( y(0) = 2 \)

Contributors

  • Integrated by Justin Marshall.