We deliver solutions for the AI eraâcombining symbolic computation, data-driven insights and deep technical expertise
DSolve can be used for finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.
The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations, and is similar for IVPs and for BVPs.
IVPs and BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear problems. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, an IVP naturally breaks up into simpler IVPs over the regions in which the coefficients are continuous.
Linear IVPs and BVPsTo begin, consider an initial value problem for a linear first-order ODE.
This is a linear first-order ODE: Notice that the general solution is a linear function of the arbitrary constant C[1]: This finds a particular solution for the initial condition : This verifies that the solution satisfies both the equation and the initial condition: Here is the solution to the same problem with the general initial condition : This verifies that the solution satisfies the equation and the initial conditions: Here is a plot of the solution: To get more information about the solutions for the problem, set : Here is a BVP for an inhomogeneous linear second-order equation:It should be noted that, in contrast to initial value problems, there are no general existence or uniqueness theorems when boundary values are prescribed, and there may be no solution in some cases.
The previous discussion of linear equations generalizes to the case of higher-order linear ODEs and linear systems of ODEs.
Here is the solution to an Initial Value Problem (IVP) for a linear ODE of order four: This verifies the solution and the initial conditions: Since this is a fourth-order ODE, four independent conditions must be specified to find a particular solution for an IVP. If there is an insufficient number of conditions, the solution returned by DSolve may contain some of the arbitrary parameters, as follows: Finally, here is the solution of an IVP for a linear system of ODEs: This verifies that the solution satisfies the system and the initial conditions: Nonlinear IVPs and BVPsMany real-world applications require the solution of IVPs and BVPs for nonlinear ODEs. For example, consider the logistic equation, which occurs in population dynamics.
This is the logistic equation: The right-hand side of the equation can be expanded to a quadratic polynomial in y[t]. Hence, the logistic equation is simply a Riccati equation, and its general solution can be easily found: This sets the intrinsic growth rate r to 1/2 and the saturation level K to 4 and solves the initial value problem: This solves the initial value problem for the logistic equation with symbolic parameters r and K: This verifies that the solution satisfies the equation and the initial condition: Here is a plot of the solution for different values of r and K: Here is an example of an IVP for a second-order nonlinear ODE whose general solution can be obtained in explicit form: This verifies that the solution satisfies the equation and the initial conditions: Finally, here is a boundary value problem for a nonlinear second-order ODE. The solution is required to satisfy boundary conditions at 0 and infinity. The Solve::ifun message is generated while finding the general solution in terms of JacobiSN, the inverse of EllipticF. The DSolve::bvlim messages are given because the limit required for satisfying the condition y′[Infinity]0 cannot be calculated for either branch of the general solution. However, the solution for the boundary value problem is found using an alternative method to determine the values of the constants C[1] and C[2] in the general solution:It may not always be possible to obtain a symbolic solution to an IVP or BVP for a nonlinear equation. Numerical methods may be necessary in such cases.
IVPs with Piecewise CoefficientsThe differential equations that arise in modern applications often have discontinuous coefficients. DSolve can handle a wide variety of such ODEs with piecewise coefficients. Some of the functions used in these equations are UnitStep, Max, Min, Sign, and Abs. These functions and combinations of them can be converted into Piecewise objects.
This converts the given expression into a Piecewise expression: Here is the general solution to a first-order ODE that contains UnitStep: Here is the solution to the same ODE with an initial condition: The solution can be plotted in the usual way. Note that the solution is continuous but not differentiable at : This verifies the solution: Here is a piecewise ODE that has Max in its coefficients:A piecewise ODE can be thought of as a collection of ODEs over disjoint intervals such that the expressions for the coefficients and the boundary conditions change from one interval to another. Thus, different intervals have different solutions, and the final solution for the ODE is obtained by patching together the solutions over the different intervals.
If there are a large number of discontinuities in a problem, it is convenient to use Piecewise directly in the formulation of the problem.
This second-order ODE contains a Piecewise term: This ODE contains the Clip function. The solutions are given in terms of Airy functions:RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4