# Differential Equations Comprehensive Exam Reviewer 4 of 4

Jessyl: Sir, paano po ba mag-find ng phase plot ng differential equation sa Wolfram Alpha?

Sir June: Umm, Jessyl, para magawa ‘yun, you can use ‘plot dy/dx phase space’. This command tells Wolfram Alpha na gusto mong makita yung phase plot, na nagpapakita ng behavior ng system mo.

Keith: Sir, ehh, how do we check kung exact yung differential equation sa Wolfram Alpha?

Sir June: Ah, good question, Keith. You’d use ‘dy/dx, test for exactness’. It’s like asking Wolfram Alpha, “Pwede ba, check mo naman kung exact itong equation ko?”

Ina: Sir, ahh, paano po kung kailangan naming maghanap ng integrating factor for a differential equation sa Wolfram Alpha?

Sir June: Ina, for that, you would type ‘dy/dx, find integrating factor’. Para bang sinasabi mo sa Wolfram Alpha, “Please, help me find the integrating factor for this equation.”

Deza: Sir, umm, paano naman po mag-separate ng variables ng differential equation using Wolfram Alpha?

Sir June: Deza, you can input ‘dy/dx, separate variables’. It’s like instructing Wolfram Alpha, “Hi, please separate the variables for me. Gusto ko silang magkahiwalay para mas madaling ma-solve.”

John Mark: Sir, ehh, how about linearizing a differential equation around an equilibrium point in Wolfram Alpha?

Sir June: John Mark, for that, you’d go with ‘dy/dx linearization around point’. It’s like saying, “Wolfram Alpha, pakisimplify naman itong equation ko around this specific point.”

Haeasa: Sir, how does Wolfram Alpha interpret the input y” + y = 0 when solving differential equations?

Sir June: Ah, Haeasa, y” + y = 0 is interpreted as a second-order linear differential equation. It’s like telling Wolfram Alpha, “Hey, we’re dealing with a second-order equation here, ha, not just any simple equation.”

Ina: Sir, what command would you use to solve a nonhomogeneous differential equation like y” + 2y’ + y = x in Wolfram Alpha?

Sir June: Ina, you can simply input the equation as ‘differential equation y” + 2y’ + y = x’. Wolfram Alpha understands that you’re dealing with a nonhomogeneous equation based on the structure of the input.

Jessyl: Sir, if we have a differential equation with a discontinuous forcing function, what function should we include in our Wolfram Alpha query?

Sir June: Jessyl, for that scenario, you’d include the Heaviside function. It’s perfect for dealing with discontinuities in your differential equations.

Keith: Sir, paano po ba specify ang boundary condition like y(1)=0 for the differential equation dy/dx = 3y in Wolfram Alpha?

Sir June: Keith, you can input it as ‘solve dy/dx = 3y with y(1)=0’. It’s like giving Wolfram Alpha the full picture: “Here’s my equation, and by the way, ito yung boundary condition.”

Deza: Sir, if we want to find an approximate solution to a differential equation at a specific point using Wolfram Alpha, which keyword would we include?

Sir June: Deza, you’d use the keyword ‘NDSolve’. It tells Wolfram Alpha, “Okay, I need a numerical or approximate solution for this point. Can you do that?”

John Mark: Sir, how about obtaining a parametric solution of a differential equation in Wolfram Alpha? Ano pong syntax ang appropriate?

Sir June: John Mark, you’d use ‘solve dy/dx parametrically’. It’s like instructing Wolfram Alpha, “I need a parametric solution. Paki-arrange naman.”

Haeasa: Sir, what would we enter in Wolfram Alpha to solve a coupled system of first-order differential equations?

Sir June: Haeasa, you can enter ‘solve coupled system dy/dx, dz/dx’. It’s like saying, “Wolfram Alpha, I have these two equations that are interconnected. Help me solve them together, please.”

Ina: Sir, which Wolfram Alpha query would we use to solve a differential equation using separation of variables?

Sir June: Ina, you’d use ‘separate variables dy/dx = x/y’. It’s a clear command telling Wolfram Alpha na gusto mong i-separate yung variables para mas madali mong ma-solve yung equation.

Jessyl: Sir, how do you instruct Wolfram Alpha to provide a solution to a differential equation that includes a definite integral?

Sir June: Jessyl, you’d type ‘solve dy/dx with integral’. It’s like telling Wolfram Alpha, “Paki-solve naman itong equation na may kasamang integral.”

Keith: Sir, when seeking a power series solution to a differential equation in Wolfram Alpha, what is the correct approach?

Sir June: Keith, for a power series solution, you can use ‘solve dy/dx as power series’. It’s like asking Wolfram Alpha, “Can you express the solution as a power series for me?”

Deza: Sir, which one would you use in Wolfram Alpha to find the Wronskian of two solutions to a differential equation?

Sir June: Deza, you’d use ‘calculate Wronskian for dy/dx, dz/dx’. This tells Wolfram Alpha na you’re looking for the Wronskian, which is a measure of the solutions’ independence.

John Mark: Sir, to solve an exact differential equation in Wolfram Alpha, which command is appropriate?

Sir June: John Mark, you can use ‘exact differential dy/dx’. It’s like saying to Wolfram Alpha, “This is an exact differential equation. Paki-solve naman.”

Haeasa: Sir, how can Wolfram Alpha be utilized to find a bifurcation point in a differential equation?

Sir June: Haeasa, you could enter ‘find bifurcation dy/dx’. This command is asking Wolfram Alpha to pinpoint where the bifurcation, or the qualitative change in behavior, occurs in the differential equation.

Ina: Sir, what query would you enter to calculate the eigenvalues of a system of differential equations in Wolfram Alpha?

Sir June: Ina, you’d use ‘calculate eigenvalues for dy/dx, dz/dx’. It’s telling Wolfram Alpha that you’re interested in the eigenvalues, which are crucial for understanding the system’s behavior.

Jessyl: Sir, if you want a qualitative analysis of a differential equation’s solution, what would you type in Wolfram Alpha?

Sir June: Jessyl, you can use ‘qualitative analysis dy/dx’. It’s a way of asking Wolfram Alpha to look beyond the numerical solutions and give you insights into the behavior and characteristics of the solutions.

Keith: Sir, when you want to find the orthogonal trajectories of a family of curves represented by a differential equation in Wolfram Alpha, you would enter what?

Sir June: Keith, you would enter ‘orthogonal trajectories of dy/dx’. It’s like instructing Wolfram Alpha, “Show me the paths that cross my given family of curves at right angles.”

Deza: Sir, to explore the long-term behavior of solutions to a differential equation using Wolfram Alpha, which keyword is important?

Sir June: Deza, you’d look for ‘asymptotic behavior dy/dx’. This asks Wolfram Alpha to analyze how the solutions behave over an extended period, giving insights into their long-term trends.

John Mark: Sir, how do you command Wolfram Alpha to perform a partial fraction decomposition of the solution to a differential equation?

Sir June: John Mark, you’d use ‘dy/dx partial fractions’. It’s a straightforward way of telling Wolfram Alpha na you want the solution in terms of partial fractions, which can simplify complex rational expressions.

Haeasa: Sir, if interested in the slope field of a differential equation, what would you input in Wolfram Alpha?

Sir June: Haeasa, you’d input ‘slope field of dy/dx’. This command is like saying, “Let’s visualize how the slope of the solution behaves across different points.”

Ina: Lastly, Sir, to find a periodic solution of a differential equation using Wolfram Alpha, the correct input would be what?

Sir June: Ina, you’d use ‘periodic solution dy/dx’. It’s a way of telling Wolfram Alpha na you’re looking for a solution that repeats itself after a certain interval, typical for periodic systems.

Sir June: And that wraps up our list! We’ve tackled a wide range of functionalities in Wolfram Alpha, from plotting phase spaces to analyzing the long-term behavior of solutions. Remember, the more you play around with these commands, the more familiar you’ll get. Don’t hesitate to explore and try them out on Wolfram Alpha. Practice makes perfect! Great job today, everyone!