Tag Archives: Elementary Differential Equations

Header Photos for Elementary Differential Equations Chapter 9 Special Second Ordered Equations

Chapter 9: Special Second-Ordered Equations

Supplementary Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

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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.

Header Photo for Elementary Differential Equations

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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