-b.1.3.1 Determine order x^2 \frac{d^2y}{dx^2 }+x\d{y}{x}+2y=\sin\left({x}\right) is linear

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In summary, "order" in this equation refers to the highest power of the independent variable (x) that appears in the equation. This equation is nonlinear because the dependent variable (y) and its derivatives are not directly proportional to the independent variable (x). To solve this equation, methods such as variation of parameters, undetermined coefficients, or Laplace transforms can be used, as well as numerical methods for approximation. The physical significance of this equation lies in its use for modeling systems that involve acceleration or vibrations in physics and engineering, such as a spring-mass system or an electric circuit with an oscillating current. Additionally, this equation can be applied to real-life situations, such as analyzing the behavior of a pendulum or a damped
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Determine order and if the equation is linear
$x^2 \frac{d^2y}{dx^2 }+x\d{y}{x}+2y=\sin\left({x}\right)$

I found this but not sure how it works

To be linear...

1- The dependent variable y and its derivatives occur to the first degree only.
2- No products of y and/or any of its derivatives appear in the equation.
3- No transcendental functions of y and/or its derivatives occur.

note just noticed the book uses t instead of x
 
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This equation is not linear since it contains a transcendental function (sin x). The order of the equation is second order.
 

FAQ: -b.1.3.1 Determine order x^2 \frac{d^2y}{dx^2 }+x\d{y}{x}+2y=\sin\left({x}\right) is linear

What is the meaning of "order" in this equation?

"Order" in this equation refers to the highest power of the independent variable (x) that appears in the equation. In this case, it is a second-order differential equation because the highest power of x is 2.

Is this equation linear or nonlinear?

This equation is nonlinear because the dependent variable (y) and its derivatives are not directly proportional to the independent variable (x).

How do I solve this equation?

To solve this equation, you can use methods such as variation of parameters, undetermined coefficients, or Laplace transforms. It is also possible to use numerical methods to approximate a solution.

What is the physical significance of this equation?

This type of equation is commonly used in physics and engineering to model systems that involve acceleration or vibrations, such as a spring-mass system or an electric circuit with an oscillating current.

Can this equation be applied to real-life situations?

Yes, this equation can be applied to various real-life situations where the behavior of a system can be described by second-order differential equations. For example, it can be used to analyze the motion of a pendulum or the behavior of a damped harmonic oscillator.

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