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Solve the differential equation.$ xy' - 2y = x^2, x > 0 $

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$$y=x^{2}(\ln x+C)$$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 5

Linear Equations

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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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Solve the differential equ…

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Solve the given differenti…

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In order to solve this equation, we want to get into the form y prime plus y p of X is Q backs as listed in the textbook and order. To this, we must divide both sides by X to get wide prime minus two. Why over acts is acts. Now we know we must multiply each of these terms by the constant of integration. So each the integral is night of negative to over Axe de Axe gives us eat the negative to natural law of acts which gives us eat the natural of X to the minus two, which is acts to the minus two because again, each the natural log simply gives us once This can literally just be crossed off. Okay, Now that we have this, we know we're gonna be multiplying the integrating factor by each of our subsequent terms. Because then were we able to integrate the right side So you can see we're just multiplying each other charms by the integrating factor. Okay, cool. Now that we've got this, we know the left hand side remains. Why axe to the negative too? However, the right hand side we now need to integrate DEA vacs over acts which is the same thing is integrating one over ox, which is natural log of X plus C. Now, lastly, for a last step, we know we must divide both sides in order to get why by itself. So we're dividing each of these terms by X to the negative too. So we end up with why is X squared times natural log of acts plus see are integrating factor.

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