**Wronskian, Exact Equations, and Bifurcation Points**

To calculate the Wronskian of solutions to a differential equation, use the command “calculate Wronskian for dy/dx, dz/dx”. For solving exact differential equations, the phrase “exact solution dy/dx” is appropriate. To find bifurcation points, the command “find bifurcation dy/dx” is used. These commands are essential for analyzing the uniqueness and behavior of differential equation solutions.

**Eigenvalues, Qualitative Analysis, and Orthogonal Trajectories**

When calculating eigenvalues of a system of differential equations, use “eigenvalues of system dy/dx, dz/dx”. For a qualitative analysis of a differential equation’s solution, “qualitative analysis dy/dx” is the correct phrase. To find orthogonal trajectories of a family of curves represented by a differential equation, use “orthogonal trajectories of dy/dx”.

**Long-term Behavior, Partial Fractions, and Slope Fields**

For exploring long-term behavior of solutions, “asymptotic behavior dy/dx” is suitable. For partial fraction decomposition of a solution, use “partial fraction decomposition dy/dx”. To visualize the slope field of a differential equation, the command “slope field of dy/dx” is appropriate.

**Periodic Solutions and Higher-Order Equations**

To find periodic solutions, use “periodic solution dy/dx”. For higher-order differential equations, like a third-order equation, the syntax “solve y”’ = x^3″ is correct.

**Specialized Equations (Logistic, Riccati, Linear Systems)**

For solving logistic differential equations, “solve logistic dy/dx = r*y*(1 – y/K)” is used. Riccati differential equations can be solved with “solve dy/dx = y^2 + x Riccati”. For systems of linear differential equations with constant coefficients, the phrase “solve system dy/dx, dz/dx constant coefficients” is suitable.

**Greens Function and Undetermined Coefficients**

To find the Green’s Function of a linear differential operator, use “find Greens Function for L”. For applying the method of undetermined coefficients, the command is “solve dy/dx undetermined coefficients”.

**Numerical Methods and Critical Point Analysis**

For stiff differential equations, “solve stiff dy/dx numerically” is appropriate. For analyzing the behavior of a system of nonlinear differential equations at critical points, the input should be “dy/dx, dz/dx critical points analysis”.

**Jacobian Determinants and Parametric Solutions**

To calculate the determinant of the Jacobian matrix for a system of differential equations, use the command “calculate Jacobian determinant for dy/dx, dz/dx”. This helps in understanding the stability and local behavior of dynamical systems modeled by differential equations.

**Parametric Solutions and Laplace Transforms**

For finding a parametric solution of a system of differential equations, the format “dy/dx, dz/dx parametric solution” is correct. This is particularly useful in situations where solutions are best expressed in terms of a third variable (the parameter). The Laplace transform, a powerful tool for solving linear differential equations, especially with initial conditions, can be computed in Wolfram Alpha using “Laplace transform of dy/dx”.

**Inverse Laplace Transforms and Existence-Uniqueness**

The inverse Laplace transform, which is crucial in solving differential equations in the s-domain back to the time domain, can be obtained with “inverse Laplace transform of F(s)”. To check the existence and uniqueness of a solution for a differential equation, which is fundamental in ensuring that a problem has a well-defined solution, use “existence and uniqueness dy/dx”.

**Nonlinear Systems and Numerical Solutions**

For nonlinear systems, the exact solution can be explored using “solve nonlinear system exactly dy/dx, dz/dx”. Numerical solutions, essential when exact solutions are not feasible, can be found using “solve dy/dx and dz/dx numerically”. This approach is crucial for complex or highly nonlinear systems where analytical solutions are not available.

**Runge-Kutta Methods**

For more complex differential equations where standard methods might not suffice, the Runge-Kutta methods (notably the fourth-order method, RK4) can be invoked with commands like “solve dy/dx Runge-Kutta”. These methods are highly regarded for their accuracy and stability in numerical solutions.