Practice question · Put in order
Order the steps for solving a first-order separable differential equation with an initial condition.
- Write the equation in the form dy/dx = g(x) h(y)
- Substitute the initial condition and solve for the constant
- Rearrange to express y explicitly, giving the general solution
- Integrate both sides, introducing a single constant of integration
- Separate the variables so that all y-terms sit with dy and all x-terms with dx
Hints
- Separation must happen before either side can be integrated.
- The initial condition can only be used once a constant exists to be determined.
Show the answer
- Write the equation in the form dy/dx = g(x) h(y)
- Separate the variables so that all y-terms sit with dy and all x-terms with dx
- Integrate both sides, introducing a single constant of integration
- Rearrange to express y explicitly, giving the general solution
- Substitute the initial condition and solve for the constant
Why
Separate, integrate, solve, then apply the condition. Using the initial condition too early, or omitting + C when integrating, both destroy the method: without the constant there is nothing for the condition to fix.
Practise Introduction to Differential Equations
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