Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. Before: a discrete sequence of hidden layers. After: the derivative of the hidden state. Traditional methods: resid…
(Newton 1671, “Problema II, Solutio particulare”). Solve the total differential equation $$3x^2-2ax+ay-3y^2y'+axy'=0.$$Solve:We have $$y'(3y^2-ax)=3x^2-2ax+ay.$$So$$dy(3y^2-ax)=(3x^2-2ax+ay)dx.$$So $$y^{3}-axy=x^3-ax^2+axy+C$$where $c$ is a constant.…
Solve equation $y'=1-3x+y+x^2+xy$ with another initial value $y(0)=1$. Solve: We solve this by using Newton's extraordinary method.We assume that the solution is analytic,which means it can be expanded in Taylor series.$y(0)=1$ means that $$ y'(0)=2…
1.Introduction 2.First-order Differential Equations Exercise2.1. Find solutons of the following intial-value problems in $\bbR^2$: (1)$2u_y-u_x+xu=0$ with $u(x,0)=2xe^{x^2/2}$; (2)$u_y+(1+x^2)u_x-u=0$ with $u(x,0)=\arctan x$. Solution: (1)Since $(-1,…
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