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First Order Differential Equations

These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

Degree and Order

Which of the following differential equations is a linear equation of order \(3:\) \[\begin{align} &(a) \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}\frac{dy}{dx}+y=5x\\ &(b) \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+y^2=x^2+3x\\ &(c) x\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}=e^x\\ &(d) \frac{d^2y}{dx^2}+\frac{dy}{dx}=\log{x}? \end{align}\]

If \(a\) and \(b\) are the order and the degree, respectively, of the differential equation \[\left(\frac{d^{4}y}{dx^{4}}\right)^{8}-\left(\frac{dy}{dx}\right)^{25}+15=0,\] what is the value of \(b-a?\)

What is the degree of the differential equation \[\frac{d^3y}{dx^3}+12\left(\frac{d^2y}{dx^2}\right)^2=x^3\log\frac{d^2y}{dx^2} ?\]

What is the degree of the following ordinary differential equation: \[k(y'')^{5}=(1+(y'')^{4})^{4}?\]

Which of the following differential equations has degree \(1 ?\) \[\begin{align} &\text{(a) } x^3\frac{d^2y}{dx^2}+(x+x^2)\left(\frac{dy}{dx}\right)^2+e^xy^3=2\sin x \\ &\text{(b) } y=2\frac{dy}{dx}+5\sqrt{1-\left(\frac{dy}{dx}\right)^2} \\ &\text{(c) } \sqrt{\frac{dy}{dx}+y}=x+4 \end{align}\]

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