The World of Separable Differential Equations
Separable differential equations are a type of first-order ODE where the variables can be separated on either side of the equation. By rearranging, all terms with one variable go on one side, and all terms with the other variable on the opposite side, allowing for straightforward integration and solution.
[include_netrun_products_block from-products="product/ged-math-workbook-comprehensive-math-practices-and-solutions-the-ultimate-test-prep-book-with-two-full-length-practice-tests/" product-list-class="bundle-products float-left" product-item-class="float-left" product-item-image-container-class="p-0 float-left" product-item-image-container-size="col-2" product-item-image-container-custom-style="" product-item-container-size="" product-item-add-to-cart-class="btn-accent btn-purchase-ajax" product-item-button-custom-url="{url}/?ajax-add-to-cart={id}" product-item-button-custom-url-if-not-salable="{productUrl} product-item-container-class="" product-item-element-order="image,title,purchase,price" product-item-title-size="" product-item-title-wrapper-size="col-10" product-item-title-tag="h3" product-item-title-class="mt-0" product-item-title-wrapper-class="float-left pr-0" product-item-price-size="" product-item-purchase-size="" product-item-purchase-wrapper-size="" product-item-price-wrapper-class="pr-0 float-left" product-item-price-wrapper-size="col-10" product-item-read-more-text="" product-item-add-to-cart-text="" product-item-add-to-cart-custom-attribute="title='Purchase this book with single click'" product-item-thumbnail-size="290-380" show-details="false" show-excerpt="false" paginate="false" lazy-load="true"] [include_netrun_products_block from-products="product/ged-math-workbook-comprehensive-math-practices-and-solutions-the-ultimate-test-prep-book-with-two-full-length-practice-tests/" product-list-class="bundle-products float-left" product-item-class="float-left" product-item-image-container-class="p-0 float-left" product-item-image-container-size="col-2" product-item-image-container-custom-style="" product-item-container-size="" product-item-add-to-cart-class="btn-accent btn-purchase-ajax" product-item-button-custom-url="{url}/?ajax-add-to-cart={id}" product-item-button-custom-url-if-not-salable="{productUrl} product-item-container-class="" product-item-element-order="image,title,purchase,price" product-item-title-size="" product-item-title-wrapper-size="col-10" product-item-title-tag="h3" product-item-title-class="mt-0" product-item-title-wrapper-class="float-left pr-0" product-item-price-size="" product-item-purchase-size="" product-item-purchase-wrapper-size="" product-item-price-wrapper-class="pr-0 float-left" product-item-price-wrapper-size="col-10" product-item-read-more-text="" product-item-add-to-cart-text="" product-item-add-to-cart-custom-attribute="title='Purchase this book with single click'" product-item-thumbnail-size="290-380" show-details="false" show-excerpt="false" paginate="false" lazy-load="true"]
Solving separable differential equations:
To solve a separable differential equation, the variables \( y \) and \( x \) are first separated. This is followed by integrating both sides of the equation independently. After integration, an arbitrary constant is introduced, and the equation is rearranged to derive the general solution, expressing the relationship between \( y \), \( x \), and the constant \( C \).
Here is an example:
\( \text{Problem: Solve the separable differential equation }\)
\(\frac{dy}{dx} = \frac{y^2 \sin(x)}{e^y}, \text{ where } y \neq 0.\)
\(\text{First, separate the variables } y \text{ and } x:\)
\(e^y dy = y^2 \sin(x) dx.\)
\(\text{Next, integrate both sides of the equation:}\)
\(\int e^y dy = \int y^2 \sin(x) dx.\)
\(\text{Perform the integrations:}\)
\(e^y = -y^2 \cos(x) + g(y),\)
\(\text{where } g(y) \text{ is an arbitrary function of } y.\)
\(\text{Since the left side is a function of } y \text{ only, } g(y) \text{ must be a constant:}\)
\(e^y = -y^2 \cos(x) + C.\)
\(\text{The general solution to the differential equation is:}\)
\(e^y + y^2 \cos(x) = C,\)
\(\text{where } C \text{ is an arbitrary constant.} \)
Here is another problem:
\( \text{Problem: Solve the differential equation }\)
\(\frac{dy}{dx} = \frac{\cos(x) + \sin(y)}{y(1 + x^2)}, \text{ with } y \neq 0 \text{ and } x \neq 0.\)
\(\text{Integrate both sides:}\)
\(\int y(1 + x^2) dy = \int (\cos(x) + \sin(y)) dx.\)
\(\text{Performing the integrations:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{where } C \text{ is an integration constant.}\)
\(\text{The general solution to the differential equation is:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{expressing a relationship between } y, x, \text{ and } C. \)
Related to This Article
More math articles
- How to Find Independent and Dependent Variables in Tables and Graphs?
- How to Use Grids to Multiply One-digit Numbers By Teen Numbers
- How to Use Comparison test for Convergence
- 7th Grade Common Core Math FREE Sample Practice Questions
- How to Graph an Equation in the Standard Form?
- How To Get A Perfect Score Of 36 On The ACT® Math Test?
- 7th Grade SOL Math Worksheets: FREE & Printable
- How to Prepare for the PSAT Math Test?
- Everything you Need to know about Limits
- Subtracting 2-Digit Numbers
What people say about "The World of Separable Differential Equations - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.