Contents
What does it mean to solve an ODE?
ordinary differential equation
An ordinary differential equation (ODE) is an equation that involves some ordinary derivatives (as opposed to partial derivatives) of a function. Often, our goal is to solve an ODE, i.e., determine what function or functions satisfy the equation. In general, solving an ODE is more complicated than simple integration.
How do you find the solution of an ODE?
Solve the ODE with initial condition: dydx=7y2x3y(2)=3. Solution: We multiply both sides of the ODE by dx, divide both sides by y2, and integrate: ∫y−2dy=∫7x3dx−y−1=74×4+Cy=−174×4+C. The general solution is y(x)=−174×4+C. Verify the solution: dydx=ddx(−174×4+C)=7×3(74×4+C)2.
Which is the method of differential equation of second order?
where P(x), Q(x) and f(x) are functions of x, by using: Undetermined Coefficients which only works when f(x) is a polynomial, exponential, sine, cosine or a linear combination of those. Variation of Parameters which is a little messier but works on a wider range of functions.
Which is the correct solution to the Ode?
Since y 1 is a solution of dy/dt + t 2 y = 0, then dy 1 /dt + t 2 y 1 = 0, or dy 1 /dt = -t 2 y 1 . What do you get if you replace dy 1 /dt by -t 2 y 1 in the equation above, and also do the same thing for y 2, since it is also a solution of the ODE?
Is the sum the same as the Ode?
the sum is another function of the same form, so it is also a solution of the ODE. To get a unique solution to the ODE, you must add an additional requirement: an initial value. You specify a value t, t 0, and you require that y (t 0) = y 0.
Which is the explicit method for an ode?
One way to pose this question is to determine how close the computed values are to the analytic solution, which we might write as . The simplest method for producing a numerical solution of an ODE is known as Euler’s explicit method, or the forward Euler method.
How to find a solution to an inhomogeneous ODE?
3. Solution to Inhomogeneous DE’s Using Integrating Factors We start with the integrating factors formula: . the general solution to the inhomogeneous first order linear ODE (1) ( x + p(t)x = q(t)) is 1 x(t) = u(t) u(t)q(t)dt + C , where u(t) = e p(t) dt. (5) 2