User:Egm6321.f09.TA/Homework statement/HW2

Homework Assignment #2 - due Wednesday, 9/23, 21:00 UTC Problem #1 ([[media:Egm6321.f09.p7-1.png|7-1]]): Complete the details of case 2, when $$h_{x}N=0$$ to obtain $$h(y)$$ Problem #2 ([[media:Egm6321.f09.p8-3.png|8-3]]): Show that the solution of $$ y'+\frac{1}{x}y=x^{2} $$ is $$y=\frac{x^{3}}{4}+\frac{c}{x}$$. Problem #3 ([[media:Egm6321.f09.p9-2.png|9-2]]): Show that  the L1_ODE_VC $$ \frac{1}{2}x^{2}y'+[x^{4}y+10]= 0 $$  is exact. Problem #4 ([[media:Egm6321.f09.p9-3.png|9-3]]): Show that $$ (\frac{1}{3}x^{3})(y^{4})y'+(5x^{3}+2)(\frac{1}{5}y^{5})=0$$ is an exact nonlinear, first order ODE. Problem #5 ([[media:Egm6321.f09.p10-3.png|10-3]]): Show that the second exactness condition for $$xyy''+x(y')^{2}+yy'=0$$ is satisfied. Problem #6 (11-2): Derive eq. 5 on p.([[media:Egm6321.f09.p10-2.png|10-2]]) by differentiating eq. 3 on p.([[media:Egm6321.f09.p10-1.png|10-1]]) with respect to $$p=y' $$. Problem #7 ([[media:Egm6321.f09.p12-1.png|12-1]]): Use $$\phi_{xy}=\phi_{yx}$$ to obtain eq. 4 on p.([[media:Egm6321.f09.p10-2.png|10-2]]). Problem #8 ([[media:Egm6321.f09.p12-2.png|12-2]]): Verify exactness condition 2, equations 4&5 on p.([[media:Egm6321.f09.p10-2.png|10-2]]). for the differential equation $$8x^{5}y'y''+2x^{2}y'+20x^{4}(y')^{2}+4xy=0$$ Problem #9 ([[media:Egm6321.f09.p12-3.png|12-3]]): Verify the exactness of the ODE $$\sqrt{x}y''+2xy'+3y=0$$. Back to HW page Back to TA page