Problem

Solve the given differential equation. $$\frac{d …

Problem 47 Hard Difficulty

Solve the given differential equation.
$$\frac{d y}{d x}-\frac{1}{2 x \ln x} y=2 x y^{3}$$

Answer

$y(x)=\sqrt{\frac{\ln x}{x^{2}(1-2 \ln x)+c_{1}}}$

Related Courses

Calculus 2 / BC

Differential Equations and Linear Algebra

Chapter 1

First-Order Differential Equations

Section 8

Change of Variables

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Video Transcript

So for this problem, we have a differential equation written on the top. Left on This is a new equation where any pulls free and the functions p of X and Q of X or is written on the right hand side. As a new equation, we can make the transformation new equals y to the power of one minus. And one monastery is just minus tape by Equation 1.8, Week 12 from the textbook. This transforms the differential equation into minus half utx minus 1/2 x l n of x times You not sequel to two X. We can multiply fruit by minus two to give us this differential equation in the form B u D x plus one over X Ellen of X, I'm she, and that's equal to minus four x. This is a differential equation, which we can solve using the integrating factor method. So the integrating factor we want to take the exponential of the integral won't go. The x Ellen X, the X So let's look at this integral in more detail. So the integral of one over X So it's natural. Log of X, the X we're gonna sold by substitution. So let us make the substitution. T equals natural log of X, then DT equals one over x times dx. This transforms the integral into an inter rule in terms of tea, and this central is simply won't over tea et on this has solution natural log t But if we substitute back t in equals natural Lagerback's This solution is the natural log of the natural log of X. So we have a double natural logarithms to go back to the left hand side of our integrating fact there. This means integrating fact that is the exponential of the natural log of the natural log of X on the first exponential, the first natural or councils with the exponential. So the integrating fact that is simply the natural log of X going back to our differential equation. This means that we can rewrite it as the direct extra votive you've times natural log of X is equal tu minus full X. It's natural logarithms of X, so the next step is to integrate both sides. Now the integral left is simple, but the integral on the right will look at in more detail because we're gonna have to integrate by parts. This is minus four times the integral of x Ellen x dx. So we take a to be equal to the natural log of X andI db the X to be equal to X the corresponding derivative of a is simply one over X andI b is equal. Do X squared over two by simple integration and differentiation. Using the integration by parts formula. This means that are integral becomes minus four times a times B. So this is minus four times X squared over two. So the four of the two becomes two to minus two x squared as natural vex minus the integral of de a d x times be. And that's the interval of X over to the X and we need to remember the minus four Car efficient service becomes plus four solving the second in school. We find that the entire solution for the integral is minus two x squared. There's not a lot of ex plus well x squared over four. The force cancel service just becomes us X squared then plus the integration constant c. So we now put this back into the differential equation. We find that you times the natural log of X equal two X squared minus two X squared Natural log of X Let's see work to find a solution for you on its own So we just divide fruit by the natural log of X onda We find that you is equal two x squared over the natural log of axe minus two x squared plus c over the natural log Riven vex But we don't just want the solution for you We want this in terms of why So you simply make the substitution u equals y to power of minus two on If we rewrite the right hand side of for a single fraction, we get that y to the power of a minus two is equal X squared times one minus two l and X plus a new constant C which has changed. We multiplying and such. And this is all over the natural log of X Mrs O Solution differential equation

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