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Solve the differential equation.$ t^2 \frac {dy}{dt} + 3ty = \sqrt {1 + t^2}, t > 0 $
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Calculus 2 / BC
Chapter 9
Differential Equations
Section 5
Linear Equations
Missouri State University
Baylor University
University of Nottingham
Lectures
13:37
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.
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In order to solve this equation, we must first get into the standard form. Why? Prime plus p of x times. Why is Q backs and orgy this? We must divide by t squared to get this into the standard form that I just mentioned. So as I said, I'm dividing by t squared. Okay, Now that we've got this, we know we must determine the integrating factor. Each of the integral of three over tee times D t is the same thing has eaten the three, not a log of tea which is the same thing as eat the natural of t cubed. Now we know eats the natural log is once this the same thing as t cubed. We have just determined are integrating factor. Okay, Now we must multiply both sides. In other words, all terms by the integrating factor that we just determined in order to know integrate both sides and sulfur Why we have t cubed. Why is the integral of tea times one plus she squared d t which gives us t cubed. Why is one third times one plus t squared 23 over to pussy We use the power room or to do this other words. We increased our expert by one and we then divided by the new exponents. Lastly, we only want this in terms of why not any sort of co fishing in front? That isn't one, which means we must divide by t cubed for the terms in order to derive our final solution plus c times t to the negative three which the same thing is pussy over too cute.
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