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Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x.$$y^{\prime}-(1+x)\left(1+y^{2}\right)=0$$

$\tan \left(x+\frac{x^{2}}{2}+C\right)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS

Section 2

Separation of Variables

Integrals

Integration

Integration Techniques

Differential Equations

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:45

Solve the differential equ…

01:54

02:12

03:34

02:58

01:56

03:56

06:41

Solve the given differenti…

04:43

Find the general solution …

Okay, We're given any derivative equation, and I replaced. Why Prime with do whatever the axe You want to solve this for? Why? First, I'm gonna go ahead and add that to the other side. So I have d y over DX equals one plus x times one plus. Why square? In order to separate the variables here, we need to get the exes on one side. So when you multiply both sides by and d x all remove it from the denominator on the left. Put it on top on the right. The only thing I need to do to finish that off is divide by the one plus y square. So I have d Y over one plus y squared is equal to one plus acts times the axe you get that lets integrate both sides, by definition, the integral of d y over one plus why squared is Arc Tangent of wine. Plus a constant which will call C one. The integral of one plus acts would be X plus X squared for two, plus another constant C two and we know if we subtract one constant from another, we just get a constant, and we actually could isolate the Why here? If we take the tangent of our attention, they cancel out, we get Why? So we're gonna take the tea engine of both sides, and we will end up with the statement that why is equal to the tangent of X plus X squared over two plus constant.

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