Integrating Factor

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Question 1

Which of the following cannot be solved using the integrating factor method?


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

Question 2

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

\[ y^2 = x^2 + 1 \]

\[  y = xe^{-x} \]

Question 3

Solve \[ \frac{dy}{dx} + 3y = e^{-3x} \] with y(0) = 1. Which of the following is the solution?

\[ y = e^{3x} \]
\[ y = e^{-3x} \]
\[ y = 3e^{-3x} \]

Question 4

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

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

\[ y = e^{x}(x + 1)  \]

Question 5

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

\[ y = -e^{-x^2}cos(x) \]
\[ y = e^{x^2}sinx \]
\[ y = \frac{1}{2}e^{x}cos(x)  \]

Second Order Problems Test Questions

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