Separation of Variables in Differential Equations

What is the primary goal of the separation of variables technique?

• To convert the equation into a polynomial

• To combine variables on both sides of the equation

• To isolate each variable on different sides of the equation

• To eliminate variables from the equation

What is the general form of a differential equation that can be solved by separation of variables?

• y times dy/dx equals x times dy

• y times dx equals x times dy

• y times dy/dx equals x

• y times dy/dx equals x part of the equation

In the first example, what operation is performed to isolate the y variable?

• Division by y squared

• Multiplication by y squared

• Addition of y squared

• Subtraction of y squared

What is the result of integrating y to the power of negative two?

• y to the power of zero

• y to the power of negative one divided by negative one

• y to the power of negative three divided by three

• y to the power of two divided by two

In the second example, what trigonometric function is used to rewrite the equation?

• Sine

• Cosine

• Tangent

• Cotangent

What is the anti-derivative of cotangent x?

• Natural log of absolute value of cosine x

• Natural log of absolute value of sine x

• Natural log of absolute value of cotangent x

• Natural log of absolute value of tangent x

What substitution is used to find the anti-derivative of cotangent x?

• u equals cotangent x

• u equals sine x

• u equals tangent x

• u equals cosine x

In the final example, what is the initial condition given?

• y(0) = 0

• y(1) = 0

• y(0) = 1

• y(1) = 1

What is the particular solution derived in the final example?

• y squared equals x squared

• y squared equals natural log x

• y squared equals natural log x squared

• y equals natural log x squared

What is the purpose of multiplying both sides by two in the final example?

• To balance the equation

• To simplify the equation

• To eliminate the constant

• To solve for y