User:Egm6321.f09.TA/Homework statement/HW1

Homework Assignment #1 - due Wednesday, 9/16, 21:00 UTC Problem #1 ([[media:Egm6321.f09.p2-2.png|2-2]]): Derive equations (1) and (2) on p.[[media:Egm6321.f09.p1-2.png|1-2]] $$ \frac{d}{dt}f(Y^{1}(t),t)=\frac{\partial f}{\partial s}(Y^{1}(t),t)\dot{Y}^{1}(t)+\frac{\partial f}{\partial t}(Y^{1}(t),t) $$ $$ \frac{d^{2}}{dt^{2}}f=f_{,s}(Y^{1},t)\ddot{Y}^{1}+f_{,ss}(Y^{1},t)(\dot{Y}^{1})+2f_{,st}(Y^{1},t)\dot{Y}^{1}+f_{,tt}(Y^{1},t) $$

Problem #2 ([[media:Egm6321.f09.p4-2.png|p.4-2]]): Derive Eq.(2) on [[media:Egm6321.f09.p4-2.png|p.4-2]] using int factor. $$ p(x)=Ae^{-x}+x-1 $$

Problem #3 ([[media:Egm6321.f09.p4-3.png|4-3]]): Show that the 1st order ODE is nonlinear. $$M(x,y)=2x^{2}+\sqrt{y} $$ $$N(x,y)=x^{5}y^{3}$$ $$(2x^{2}+\sqrt{y})+x^{5}y^{3}y'=0$$

Problem #4 ([[media:Egm6321.f09.p5-2.png|5-2]]): Show that $$F(x,y,y')=0$$ in Eq. (3) is a nonlinear 1st order ODE. (Hint: Define the differential operator $$D(.)$$ associated with equation (3).)

Problem #5 ([[media:Egm6321.f09.p6-1.png|6-1]]): Generating exact nonlinear 1st order ODEs: Let $$ \Phi(x,y)=6x^{4}+2y^{3/2}$$, then $$ M=\Phi_{x}$$ $$ N=\Phi_{y}$$ Complete the details and invent 3 more examples.

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