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Solve the initial-value problem.

$ xy' + y = x \ln x, y(1) = 0 $

$$

y=\frac{x}{2} \ln x-\frac{x}{4}+\frac{1}{4 x}

$$

Differential Equations

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Oregon State University

University of Nottingham

Idaho State University

we can tell from looking at the equation. Given that R p of X is one over acts are Cuba is not the work of acts. Therefore, integrating factor e to the integral of p of X eat thean trickle of one over axe dx integrate this We end up with eats the natural of acts recall each The natural log is simply one which means our interpreting factors simply acts. Therefore, multiply each of our terms by the intriguing factor and then the right hand side. We will now be integrating axe natural with ex DX. Okay, you can use integration by parts. To do this, you can use U substitution. I'm using integration by parts as you can see and I'm actually gonna have to do this a second time because it's a little bit of a complicated one with interesting by parts recall. I'm using the power rule. Increased exports by one divide by the new exponents. Now initial condition is C is 1/4 therefore y as X over to natural Kovacs minus X over four plus one over four times acts