Multiple Choice
Find the particular solution that satisfies the given initial condition .
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13. Intro to Differential Equations
Separable Differential Equations
Multiple Choice
Which of the following differential equations is separable?
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B
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D
Verified step by step guidance1
Step 1: Understand the concept of separable differential equations. A differential equation is separable if it can be written in the form \( \frac{dy}{dx} = g(x)h(y) \), where the variables \(x\) and \(y\) can be separated into two distinct functions.
Step 2: Analyze each given equation to determine if it is separable. For \( \frac{dy}{dx} = x + y \), the terms \(x\) and \(y\) are added together, making it impossible to separate them into distinct functions of \(x\) and \(y\). Therefore, this equation is not separable.
Step 3: For \( \frac{dy}{dx} = y^2 + x^2 \), the terms \(y^2\) and \(x^2\) are added together, which also prevents separation into distinct functions of \(x\) and \(y\). Thus, this equation is not separable.
Step 4: For \( \frac{dy}{dx} = xy \), the terms \(x\) and \(y\) are multiplied together, allowing the equation to be rewritten as \( \frac{1}{y} dy = x dx \). This shows that the variables can be separated, making this equation separable.
Step 5: For \( \frac{dy}{dx} = e^{x+y} \), the exponential term \(e^{x+y}\) cannot be separated into distinct functions of \(x\) and \(y\) because \(x + y\) is combined inside the exponent. Therefore, this equation is not separable.
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