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Category Archives: Differential Equations
Determining if a given Differential Equation is Separable or Not
Determine whether each of the following differential equations is or is not separable, and, if it is separable, rewrite the equation in the form dy/dx=f(x) g(y).
\qquad \textbf{a}) \quad \frac{dy}{dx}=xy-3x-2y+6
\qquad \textbf{b})\quad \frac{dy}{dx}=\sin \left( x+y \right)
\qquad \textbf{c}) \quad y\frac{dy}{dx}=e^{x-3y^2}
Solution:
Part A
\begin{align*}
\frac{dy}{dx} & = xy-3x-2y+6 \\
\frac{dy}{dx} & = \left( xy-3x \right)-\left( 2y-6 \right)\\
\frac{dy}{dx} & = x\left( y-3 \right)-2\left( y-3 \right)\\
\frac{dy}{dx} & = \left( x-2 \right)\left( y-3 \right)
\end{align*}Since F(x,y) is factorable in the form f(x) g(y), the given differential equation is separable.
Part B
\begin{align*}
\frac{dy}{dx} & = \sin\left( x+y \right) \\
\frac{dy}{dx} & = \sin\left( x \right)\cos\left( y \right) +\cos\left( x \right)\sin\left( y \right)\\
\end{align*}Since F(x,y) is not factorable in the form f(x) g(y), the given differential equation is not separable.
Part C
\begin{align*}
y \frac{dy}{dx} & = e^{x-3y^2}\\
y \frac{dy}{dx} & = \frac{e^x}{e^{3y^2}}\\
\frac{dy}{dx} & =\frac{e^x}{y e^{3y^2}} \\
\frac{dy}{dx} & = e^x \left( \frac{1}{ye^{3y^2}} \right) \\
\end{align*}Since F(x,y) is factorable in the form f(x) g(y), the given differential equation is separable.
Mathematics
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Separation of Variables| Elementary Differential Equations|dela Fuente, Feliciano, and Uy|Example 2
Elementary Differential Equations by dela Fuente, Feliciano, and Uy Problem Exercises Solution Guides
Chapter 1: Introduction: Definitions
Chapter 2: Separation of Variables
Chapter 3: Homogeneous Differential Equations
Chapter 4: Exact Differential Equations
Chapter 5: Non-Exact Differential Equations, Integrating Factors
Chapter 6: First-Ordered Linear Equations
Chapter 7: The Bernoulli Equation
Chapter 8: Differential Equations with Coefficients Linear in X and Y
Chapter 9: Special Second-Ordered Equations
Chapter 10: Applications of Ordinary First-Ordered Differential Equations
Chapter 11: Linear Equations of Higher Order
Chapter 12: The homogeneous Linear Equations with Constant Coefficients
Chapter 13: The Nonhomogeneous Linear Equations with Constant Coefficients
Chapter 14: Applications of the Higher-Ordered Linear Equations
Chapter 15: The Hyperbolic Functions
Chapter 16: The LaPlace Transforms; Gamma Functions
Chapter 17: Mathematical Series
Chapter 18: Determinants and Matrices
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