The ordinary differential equation (ODE) given is:
\[ y'' + y = e^x \]
This is a second-order linear ordinary differential equation. Here is a breakdown of the components:
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Order: The highest derivative is \( y'' \), which is the second derivative of \( y \). Therefore, it is a second-order ODE.
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Linearity: The equation is linear because \( y \) and its derivatives appear linearly (i.e., they are not multiplied together or raised to any power other than 1).
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Homogeneous vs. Non-homogeneous: If the equation were of the form \( y'' + y = 0 \), it would be homogeneous. However, because there is a non-zero term \( e^x \) on the right side, this ODE is classified as non-homogeneous.
In summary, the equation \( y'' + y = e^x \) is a second-order linear non-homogeneous ordinary differential equation.