Understanding Directly Integrable Differential Equations

What is a directly integrable differential equation?

• An equation that requires numerical methods for solutions.

• An equation that cannot be solved using integration.

• An equation where the derivative is a function of the independent variable only.

• An equation where the derivative is a function of the dependent variable only.

In the example dydx = 6x^2 + 4, what is the antiderivative of 6x^2?

• x^3

• 2x^3

• 6x^3

• 3x^3

What is the constant of integration in the general solution of dydx = 6x^2 + 4?

• No constant is needed

• C

• C2

• C1

How do you isolate y' in the equation x^2y' = -1?

• Subtract x^2 from both sides

• Divide both sides by x^2

• Multiply both sides by x^2

• Add x^2 to both sides

What is the particular solution for y' = -1/x^2 when y(1) = 3?

• y = 1/x + 3

• y = -1/x + 3

• y = 1/x + 2

• y = -1/x + 2

Which method is used to integrate x e^x in the second-order equation example?

• Trigonometric substitution

• Integration by parts

• Partial fractions

• Substitution

What is the antiderivative of e^x?

• x e^x + C

• ln|x| + C

• 1/x + C

• e^x + C

In the second-order equation example, what is the antiderivative of a constant C1?

• C1x

• C1/x

• ln|C1|

• C1

What is the particular solution for d^2y/dt^2 = cos(t) with y'(0) = 0 and y(0) = 1?

• y = cos(t) + 2

• y = -cos(t) + 1

• y = 2 - cos(t)

• y = 1 - cos(t)

What is the antiderivative of cos(t)?

• -sin(t) + C

• sin(t) + C

• cos(t) + C

• -cos(t) + C