User:Egm6321.f09.Team8/HW2

= Problem #1 =

Given
([[media:Egm6321.f09.p7-1.png|7-1]]): Complete the details of case 2, when $$h_{x}N=0$$ to obtain $$h(y)$$

Solution
[[Media:HW2, Page 1.PNG]] [[Media:HW2, Page 2.PNG]] [[Media:HW2, Page 3.PNG]]

= Problem #2 =

Given
([[media:Egm6321.f09.p8-3.png|8-3]]): Show that the solution of $$ y'+\frac{1}{x}y=x^{2} $$  is  $$y=\frac{x^{3}}{4}+\frac{c}{x}$$.

Solution
[[Media:HW2, Page 3.PNG]] [[Media:HW2, Page 4.PNG]]

= Problem #3 =

Given
([[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.

Solution
[[Media:HW2, Page 5.PNG]] [[Media:HW2, Page 6.PNG]]

= Problem #4 =

Given
([[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.

Solution
[[Media:HW2, Page 7.PNG]] [[Media:HW2, Page 8.PNG]] [[Media:HW2, Page 9.PNG]]

= Problem #5 =

Given
([[media:Egm6321.f09.p10-3.png|10-3]]): Show that the second exactness condition for $$xyy''+x(y')^{2}+yy'=0$$ is satisfied.

Solution
[[Media:HW2, Page 10.PNG]] [[Media:HW2, Page 11.PNG]]

= Problem #6 =

Given
(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 $$p=y' $$.

Solution
[[Media:HW2, Page 12.PNG]]

= Problem #7 =

Given
([[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]]).

Solution
[[Media:HW2, Page 13.PNG]] [[Media:HW2, Page 14.PNG]]

= Problem #8 =

Given
([[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]]).

Solution
[[Media:HW2, Page 14.PNG]] [[Media:HW2, Page 15.PNG]]

= Problem #9 =

Given
([[media:Egm6321.f09.p12-3.png|12-3]]): Verify the exactness of the ODE $$\sqrt{x}y''+2xy'+3y=0$$.

Solution
[[Media:HW2, Page 16.PNG]] [[Media:HW2, Page 17.PNG]]