Solving First-Order Ordinary Differential Equations

Separable First-Order ODEs

  1. Separate the variables by putting all terms with “y” on one side of the equation and all terms with “x” on the other side.
  2. Integrate both sides of the equation with respect to their respective variables.
  3. Solve for y to find the general solution, adding a constant of integration at the end of the integration step; if an initial condition is given, substitute it to find the particular solution.

Key Steps in Detail

A separable first-order ODE can be written as dy/dx = f(x)g(y), where f(x) is a function of x only and g(y) is a function of y only. Then, divide both sides of the equation by g(y), multiply both sides by dx. This will give the equation in the form dy/g(y) = f(x)dx. Then perform integration on both sides of the equations. Then, evaluate the integrals and simplify the resulting equation to isolate y. Remember to add a constant of integration (C) on one side of the equation after integration. Then, solve the differential equation dy/dx = xy.

Linear First-Order ODEs

  1. Write the equation in standard form.
  2. Find the integrating factor.
  3. Multiply the equation by the integrating factor.
  4. Integrate both sides to find the general solution.

Key Steps in Detail

Ensure the equation is in the form dy/dx + P(x)y = Q(x). Then, identify the functions P(x) and Q(x). Then, calculate the integrating factor using the formula μ(x) = e^(∫P(x)dx). Then, integrate P(x) with respect to x and exponentiate the result. Then, multiply both sides of the equation by the integrating factor μ(x). Notice that the left side should now look like a product rule derivative: d/dx (μ(x)y) = μ(x)Q(x). Then, integrate both sides of the equation with respect to x. Then, simplify and solve for “y” to get the general solution.

Exact First-Order ODEs & Integrating Factors

  1. Write the equation in standard form (express the ODE as M(x,y)dx + N(x,y)dy = 0, where M and N are functions of x and y). Make sure to check if the partial derivative of M with respect to y (∂M/∂y) equals the partial derivative of N with respect to x (∂N/∂x). If they are equal, the equation is exact and you can proceed to integration; if not, proceed to step 2 (finding an integrating factor).
  2. Find an integrating factor if necessary. If the equation is not exact, look for a function μ(x,y) such that when multiplied by the equation, it makes it exact. Check for simple cases. Sometimes μ(x,y) can be a function of only x or only y. General Case: if not, calculate μ(x,y) using the formula: (∂N/∂x – ∂M/∂y) / M (if μ is a function of x only) or (∂M/∂y – ∂N/∂x) / N (if μ is a function of y only). Then, integrate to find μ (once you have the expression for μ, integrate with respect to the appropriate variable (x or y)).
  3. Solve the exact equation. Multiply by the integrating factor (if found): Multiply the original equation by the integrating factor μ(x, y). Then, integrate M(x, y) with respect to x, treating y as a constant, and add a function of y (say, H(y)) to the result. Then, differentiate the result with respect to y and set it equal to N(x, y) to solve for H(y). The final solution will be the function F(x, y) obtained from the integration, where F(x, y) = C (C is the constant of integration).