CHAPTER 4 APPROXIMATE SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS

CHAPTER 4: APPROXIMATE SOLUTION OF NON-LINEAR DIFFERENTIAL EQUATIONS Sections 4. 1 and 4. 2 Benjamin Deaver

Agenda Section 4. 1 – Spontaneous Singularities ◦ Difference b/w Linear and Non Linear Equations & Definitions ◦ Definition of Singularities & Poles ◦ Fixed vs spontaneous Singularities ◦ Example 1, A and B Section 4. 2 – Approximate Solution of 1 st Order Non-Linear Differential Equations ◦ Infinite spontaneous singularities ◦ Example 2 ◦ Summary November 2 nd 2016 Benjamin G. P. Deaver 2

Non-Linear Differential Equation Non-linear differential equations are those where the dependent variable or its derivatives are at a power different than 1 Linear Differential equations Non-linear differential equations ◦ Y” + xy’ + 2 xy = 14 x ◦ Y” + 3 y’ + xy 2 = 0 ◦ Y” + 14 Y’ + y = 0 ◦ Y” + 3 y’y + xy = 14 ◦ Y” + xy’ = sin(x) ◦ Y” + xy’ = sin(y) ◦ Y” + y = 1/x ◦ Y” + 3 xy’ 2 = 0 Benjamin G. P. Deaver 3

Singularities and Poles ◦ Singularity points – a point at which a function f(X) is undefined or infinite, specifically where f’(x) does not exist. ◦ If a point is not a singular point it is called an ordinary point. ◦ Poles – Are a type of singularity that can be represented in the form F(z) = g(z) / (z-zo)n N is a positive integer referred to as the “order” of the pole g(z) is analytic at zo, and f(zo) ≠ 0 F(z) = (12 z – 4) / ((z-3)2(z+1)) has a 2 nd order pole at z=3 and a simple pole at z =-1 Benjamin G. P. Deaver 4

Fixed vs. Spontaneous singularities Solutions to Non-Linear Differential Equations have more complex singularities than Linear Differential Equations Fixed singularities Spontaneous or movable singularities The location of singularities are independent of the initial condition & only depend on the singularities of the coefficients. The location of singularities move around the plane dependent on initial conditions. For Example: Y’’ + p(t)y’ + g(t)y = 0, Y’’ + (1/t-1)y’ + 3 y = 0, then p(t) will influence the location of the singularities. Benjamin G. P. Deaver 5

Example 1 A: "Linear" fixed singularity ◦ Benjamin G. P. Deaver 6

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Example 1 B: "Non-Linear" spontaneous singularity Non-linear diff equation y’ = y 2; y(0)=1 This non-linear differential equation does NOT have a singularity at x=1 as demonstrated by this Slope Field diagram Benjamin G. P. Deaver 10

Solving y’ = 2 y; y(0)=1 ◦ Benjamin G. P. Deaver 11

Solving y’ = 2 y; y(0)=1 Benjamin G. P. Deaver 12

Solving y’ = 2 y; y(0)=2 Benjamin G. P. Deaver 13

Infinite Spontaneous Singularities ◦ Benjamin G. P. Deaver 14

Infinite Spontaneous Singularity Example 2: ◦ 2/12/2022 Benjamin G. P. Deaver 15

Infinite Spontaneous Singularity Example 2: Non-linear diff equation y’ = y 2 + x y(0) = 0 This diagram shows that the solution has an infinite number of singularities along the positive real axis. Looks sort of like tan(x)! Notice that as it gets larger on the X the space between the singularities gets smaller. Benjamin G. P. Deaver 16

Solving y’ = 2 y + x y(0) = 0 We are now going to have to face this BEAR of a DE here. Luckily our friend “ The Book” had a clever substitution for us to use. November 2 nd 2016 Benjamin G. P. Deaver 17

Solving y’ = 2 y +x y(0) = 0 ◦ November 2 nd 2016 Benjamin G. P. Deaver 18

Solving y’ = 2 y +x y(0) = 0 As x-> infinity November 2 nd 2016 Benjamin G. P. Deaver 19

Solving y’ = y 2 + x y(0) = 0 ◦ 2/12/2022 Benjamin G. P. Deaver 20

Summary ◦ Whew! What have we learned? ? ? ◦ First, we learned that the solutions for Non-Linear Differential Equations have UNPREDICTABLE BEHAVIORS. ◦ Next, we have learned that Non-Linear Differential Equations not only have FIXED Singularities, but they also have SPONTANEOUS or MOVABLE Singularities which are Singularities which move around the plane as the initial conditions change. ◦ Then, we learned that it is DIFFICULT to ANALYTICALLY SOLVE Non-Linear Differential Equations. ◦ Finally, we learned a few techniques for APPROXIMATING the solutions of First Order Non-Linear Differential Equations. These approximations include: ◦ Approximations at Infinity, where terms can be ignored to simplify the differential equation. ◦ Substitutions which enable the differential equation to become separable and solvable. November 2 nd 2016 Benjamin G. P. Deaver 21