Solving First-Order Ordinary Differential Equations
Separable First-Order ODEs
- Separate the variables by putting all terms with “y” on one side of the equation and all terms with “x” on the other side.
- Integrate both sides of the equation with respect to their respective variables.
- 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
- Write the equation in standard form.
- Find the integrating factor.
- Multiply the equation by the integrating factor.
- 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
- 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).
- 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)).
- 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).