Questions and Answers
The differential equation dP/dt = t/(t^2 + 1) * P is separable.
True
The solution of the differential equation in the second example involves integrating I(t) = beta * s(t) * cos(Pi/12 * (t-6)).
False
The intensity of light is proportional to the area and the cosine of the angle between the light beam direction and the vertical.
True
The angle between the light beam and the vertical is assumed to be a quadratic function of time T.
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The temperature gradient is represented by DT/Dr, and the solution is found using the given initial conditions and the constant thermal conductivity.
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The solution for the temperature at a distance R from the center is given as T(R) = Q / (4πKR + K).
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The first example in the video discusses the growth of the right area of a circular leaf.
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The inner and outer radii of the hollow iron ball have the same temperature dependence on time and distance.
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Study Notes
- The video discusses the applications of separable equations in differential equations, using two examples.
- The first example is about the growth of a population described by the differential equation dP/dt = t/t^2 + 1 * P, where P(t) is the population function at time t.
- The equation is separable, allowing the separation of variables: dP/P = t dt / t^2 + 1.
- The solution is found by integrating both sides and obtaining ln(P(t)) = 12ln(t^2 + 1) + C.
- The second example discusses the growth of the left area of a circular leaf, with the area being proportional to the left radius and the intensity of light.
- The intensity of light is proportional to the area and the cosine of the angle between the light beam direction and the vertical.
- The angle between the light beam and the vertical is assumed to be a linear function of time T.
- The solution for the intensity is obtained by integrating I(t) = beta * s(t) * cos(Pi/12 * (t-6)).
- A hollow iron ball's temperature is dependent on time and distance, with the inner and outer radii having different temperatures.
- The temperature gradient is represented by DT/Dr, and the solution is found using the given initial conditions and the constant thermal conductivity.
- The solution for the temperature at a distance R from the center is given as T(R) = Q / (4πKR + K).
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