Linear Algebra & ODEs Tutorial Sheet 8

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What is the condition for the ODE (ax + by)dx + (kx + ly)dy = 0 to be exact?

a = -k, b = -l

a = l, b = k (correct)

a = -l, b = -k

a = k, b = l

What is the solution to the ODE 2xyex dx + ex dy = 0, y(0) = 2?

y = e^(-2x)

y = e^(2x)

y = 2e^x

y = 2e^(-x) (correct)

What is the form of the ODE dy/dx + 3x^2y = x^2, y(0) = 2?

Reducible to exact

Exact

Non-linear

Linear (correct)

What is the condition for the existence of a unique solution of the IVP dy/dx = e^(2y), y(0) = 0?

The function e^(2y) is continuous and Lipschitz Signup and view all the answers

What is the solution to the ODE (1 + 2x) cos(y)dx + sec(y)dy = 0?

tan(y) = -x - x^2 + c Signup and view all the answers

Does the IVP (x - 2)dx = y; y(2) = 1 have a solution?

No, never Signup and view all the answers

How do you determine whether a first-order ODE is exact or reducible to exact?

By checking if the given ODE can be written in the form M(x,y)dx + N(x,y)dy = 0, and then verifying that (∂M/∂y) = (∂N/∂x). Signup and view all the answers

What is the significance of the integrating factor in solving linear ODEs?

The integrating factor helps in transforming a non-exact linear ODE into an exact ODE, enabling the determination of the general solution. Signup and view all the answers

When can an initial value problem (IVP) be guaranteed to have a unique solution?

An IVP is guaranteed to have a unique solution if it satisfies the conditions of the existence and uniqueness theorem, which involves the continuity and Lipschitz continuity of the function defining the ODE. Signup and view all the answers

What is the difference between a linear ODE and a reducible to linear ODE?

A linear ODE is of the form dy/dx + P(x)y = Q(x), whereas a reducible to linear ODE can be transformed into a linear ODE through a substitution or other transformation. Signup and view all the answers

How do you solve a reducible to linear ODE?

By using an appropriate substitution or transformation to convert it into a linear ODE, and then applying the standard method for solving linear ODEs. Signup and view all the answers

What is the importance of initial conditions in solving IVPs?

Initial conditions specify the value of the dependent variable at a particular point, enabling the determination of a unique solution to the IVP. Signup and view all the answers