Separation of Variables Method

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Question 1

Which of the following cannot be solved by the separation of variables method?


\[ \frac{dy}{dx} = \frac{y}{x}  \]
\[  \frac{dy}{dx} = x + y  \]

Question 2

Given \[  \frac{dy}{dx} = \frac{x}{y}  \] with initial condition y(0) = 1, find the solution to the initial value problem.


\[  y = x  \]

Question 3

Solve \[  \frac{dy}{dx} - 2y(x+1) = 0  \] with y(0) = 1. Which of the following is the solution?

\[  y = e^{x^2 + 2x} \]
\[  y = e^{x^2 + 2x} \]
\[  y = e^{x^2 + 2x} \]

Question 4

Given \[  \frac{dy}{dx} = 2x + 5 \] with initial condition y(0) = 1, find the solution to the initial value problem.

\[ y = 2x^{2} + 5 \]

\[ y = 5 - x  \]

Question 5

Solve \[ \frac{dy}{dx} = ysin(x) \] with initial condition y(0) = 1. Which of the following is the solution?

\[ y = e^{cos(x)}  \]
\[ ln(y) = 1 - cos(x) \]
\[ y = e^{sin(x) - 1} \]

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