# Differential Equations Comprehensive Exam Reviewer 3 of 4

Pumasok si Pritts sa Room.

Jessyl: Sir, ano po yung true for a system of linearly independent solutions of a differential equation?

Sir June: Jessyl, sa linearly independent solutions, hindi sila ma-eexpress as a linear combination of each other. This independence is crucial para sa diversity at uniqueness ng mga solutions.

Keith: Sir, ano yung significance ng characteristic equation sa pag-solve ng linear differential equations?

Sir June: Actually! The characteristic equation is important kasi it’s used to find the eigenvalues. These eigenvalues are key sa pag-determine ng behavior at solutions ng differential equation.

Ina: Sir, paano po magkaiba ang PDE sa ODE?

Sir June: Ina, ang PDE involves derivatives with respect to more than one variable, making it generally more complex than an ODE, na involved lang ang derivatives with respect to a single variable.

Deza: Sir, ano po ba ang ‘order’ ng differential equation?

Sir June: Deza, ‘yung ‘order’ ng differential equation, it refers to the highest derivative present in the equation. It gives you an idea kung gaano ka-complex yung equation, especially in terms of its derivatives.

John Mark: Sir, ano naman po yung typically involved sa ‘boundary value problem’ sa context ng differential equations?

Sir June: John Mark, sa ‘boundary value problem’, ang solutions are defined over an interval with specific values at the endpoints. This setup is important sa pag-ensure na yung solution fits within certain constraints or conditions.

Haeasa: Sir, saan po ba primarily used yung method of Frobenius in differential equations?

Sir June: Haeasa, yung method of Frobenius is primarily used for solving equations with singular points. It’s a powerful method na nag-aallow sa atin to find solutions near those points.

Ina: Sir, ano po yung common application ng differential equations sa engineering?

Sir June: Ina, sa engineering, one of the common applications of differential equations is in signal processing. It’s crucial sa pag-analyze at pag-understand ng various signals and systems.

Jessyl: Sir, ano po yung primary benefit ng paggamit ng eigenvalues and eigenvectors sa pag-solve ng systems of differential equations?

Sir June: Jessyl, the primary benefit is they provide a systematic method for finding general solutions. It simplifies the process at nagbibigay ng clearer understanding ng structure ng solutions.

Keith: Sir, in solving differential equations, what does ‘linearly dependent’ imply about the solutions?

Sir June: Keith, ‘pag ang solutions are ‘linearly dependent’, it means one solution can be expressed as a multiple of another. It indicates a certain relationship between the solutions na important sa pag-analyze ng behavior ng system.

Deza: Sir, ano po yung best describes a separable differential equation?

Sir June: Deza, a separable differential equation is one where variables can be separated on different sides. It makes it possible to solve each part individually, simplifying the whole solving process.

John Mark: Sir, paano po ba i-represent ang first-order differential equation sa Wolfram Alpha?

Sir June: John Mark, to represent a first-order differential equation, you would typically use the syntax like ‘dy/dx = y + x’. It’s a direct and clear way to input the equation into the system.

Haeasa: Sir, how about solving a differential equation with an initial condition in Wolfram Alpha?

Sir June: Haeasa, to solve a differential equation with an initial condition, you would type something like ‘dsolve y'(x)=x and y(0)=1’. It combines the equation itself with the initial condition, providing a complete picture for the solution.

Ina: Sir, paano po mag-ask sa Wolfram Alpha to plot the solution of a differential equation?

Sir June: Ina, to ask Wolfram Alpha to plot the solution, you can use syntax like ‘plot dy/dx = x^2’. It’s a straightforward command na nagpapakita ng visual representation ng solution.

Jessyl: And Sir, paano po correctly specify a second-order differential equation sa Wolfram Alpha?

Sir June: Jessyl, for a second-order differential equation, you’d use syntax like ‘d^2y/dx^2 = x’. It clearly indicates na you’re dealing with a second-order derivative.

Keith: Sir, how would you instruct Wolfram Alpha to solve a system of differential equations?

Sir June: Keith, to solve a system, you can use a command like ‘solve {dy/dx = x, dz/dx = y + z}’. It tells Wolfram Alpha that you’re dealing with multiple equations na need to be solved together.

Deza: Sir, which command would find a particular solution for a differential equation in Wolfram Alpha?

Sir June: Deza, to find a particular solution, you could use a command like ‘dy/dx = sin(x), y(π) = 2’. It specifies the equation and a particular point, giving you the specific solution at that point.

John Mark: Sir, if you want to solve a homogeneous differential equation using Wolfram Alpha, ano po yung keyword na gagamitin?

Sir June: John Mark, you can use a format like ‘dy/dx = x/y, solve for y’ and specify na it’s a homogeneous equation. It gives Wolfram Alpha the info it needs to approach the problem correctly.

Haeasa: Sir, to find the Laplace transform of a differential equation in Wolfram Alpha, paano po?

Sir June: Haeasa, you would type something like ‘Laplace transform of dy/dx’. It’s a direct command that tells the system exactly what you want.

Ina: Sir, how can Wolfram Alpha help in checking the stability of a differential equation’s solution?

Sir June: Ina, you can use a command like ‘stability of dy/dx’ to check the stability. It’s a way to get insights into the behavior of the solution over time.

Jessyl: Sir, what syntax in Wolfram Alpha would you use to find the numerical solution of a differential equation?

Sir June: Jessyl, you can use ‘dy/dx, find numerical solution’. It’s a straightforward command that prompts Wolfram Alpha to give you a numerical approximation of the solution.

Keith: Sir, to explore the series solution of a differential equation using Wolfram Alpha, ano po ang i-input?

Sir June: Keith, you’d input something like ‘series solution of dy/dx’. It’s a clear command that directs Wolfram Alpha to find the solution in series form.

Deza: And Sir, when looking for an implicit solution of a differential equation in Wolfram Alpha, paano po ba?

Sir June: Deza, you would type ‘solve dy/dx implicitly’. It’s a direct way to instruct Wolfram Alpha to find an implicit solution for the equation.

John Mark: Sir, for solving a Bernoulli differential equation in Wolfram Alpha, which format is correct?

Sir June: John Mark, you can use a format like ‘solve dy/dx = y^2 as Bernoulli’. It specifically tells Wolfram Alpha that you’re dealing with a Bernoulli differential equation and want it solved accordingly.

Haeasa: Sir, to obtain the Fourier series of the solution to a differential equation, ano po ang gagamitin?

Sir June: Haeasa, you’d use ‘dy/dx, compute Fourier series’. It’s a clear command that directs Wolfram Alpha to compute the Fourier series for the solution.

Ina: And lastly, Sir, if you need to apply Euler’s method to a differential equation in Wolfram Alpha, paano po?

Sir June: Ina, you would enter ‘solve dy/dx, Euler’s method’. It’s a straightforward way to tell Wolfram Alpha that you want to apply Euler’s method to solve the differential equation.

Sir June: And there we have it! Don’t worry, last na yung susunod 🙂