Textbook Question
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = y³sin t, y(0) = 1
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13. Intro to Differential Equations
Separable Differential Equations
2:09 minutesProblem 9.R.9
Textbook Question
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) = (2t+1)(y²+1)
Verified step by step guidance1
Recognize that the given differential equation is separable since it can be written as \(y'(t) = (2t+1)(y^2 + 1)\), which allows us to separate variables involving \(y\) and \(t\) on opposite sides.
Rewrite the equation in differential form as \(\frac{dy}{dt} = (2t+1)(y^2 + 1)\), then separate variables to get \(\frac{dy}{y^2 + 1} = (2t + 1) dt\).
Integrate both sides: integrate \(\int \frac{dy}{y^2 + 1}\) with respect to \(y\) and \(\int (2t + 1) dt\) with respect to \(t\).
Recall that \(\int \frac{dy}{y^2 + 1} = \arctan(y) + C_1\) and \(\int (2t + 1) dt = t^2 + t + C_2\), where \(C_1\) and \(C_2\) are constants of integration.
Combine the results to write the implicit general solution as \(\arctan(y) = t^2 + t + C\), where \(C\) is a constant that absorbs \(C_1\) and \(C_2\). Optionally, solve for \(y\) by taking the tangent of both sides to express \(y\) explicitly.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written as the product of a function of t and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.
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General Solution of Differential Equations
The general solution includes all possible solutions and typically contains an arbitrary constant from integration. It represents a family of functions satisfying the differential equation, capturing the full set of behaviors described by the equation.
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