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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.

Differential Equations - Variable Separable

What is the solution of the differential equation

$4\left(\frac{dx}{dy}\right)=8x-y^2?$

Details and assumptions

Use $$C$$ as the constant of integration.

What is the solution of the differential equation $\frac{1}{5}\left(\frac{1}{{x}^{3}}\frac{dx}{dy}-\frac{1}{{x}^{2}}y\right) =y\sin{{y}^{2}} ?$

Details and assumptions

Use $$C$$ as the constant of integration.

What is the solution of the differential equation \begin{align} (8x^2-5yx^2)\frac{dy}{dx}+(5y^3+xy^3) &=0, \\ y(1) &=1? \end{align}

What is the solution of the differential equation \begin{align} \log{\frac{dy}{dx}} &=8x+7y,\\ y(0)&=0? \end{align}

Note: In this problem, $$\log(x)$$ is the natural logarithm (log base e) of $$x.$$

What is the solution of the differential equation $\log{\frac{dy}{dx}}=5x+2y, y(0)=0?$

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