Separate, Integrate, Solve
Whenever you can herd all the y terms to one side and all the t terms to the other, solving the equation is just two integrals.
Spotting separability
An equation is separable when the right side factors into a pure function of x times a pure function of y. The equation y prime equals x times y is separable, while y prime equals x plus y is not, since that sum cannot be factored that way. Learning to see the factorization is most of the battle.
y' = g(x)\,h(y)The three-step method
First rewrite the equation so every y sits with dy and every x sits with dx. Second, integrate both sides. For y prime equals x times y, dividing by y and integrating gives the log of the absolute value of y equals x squared over 2 plus C. Third, solve for y by exponentiating to get y equals A times e to the x squared over 2.
\int \frac{dy}{y} = \int x\,dxLost solutions and checks
Dividing by h of y can silently discard constant solutions, such as y equal to 0 in the example above, so check those separately. Use the initial condition to pin down the constant A. Finally, verify by differentiating your answer; a solution must reproduce the original equation exactly.
y = A\,e^{x^2/2}Flashcard
Form of a separable differential equation
Check your understanding
Solve the initial value problem y prime equals 2y with y of 0 equal to 5.