User:Egm6321.f09.TA/Exam1grading

Exam 1 point breakdown

Consider the general nonlinear 2nd-order ODE of the form $$\sqrt{x}y''+\frac{2+x^3}{\sqrt{x}}y'+4x^{3/2}y=0$$. (1.1)  Question 1 

Consider the general nonlinear second order ODE of the form $$F(x,y,y',y'')=0$$. (1.2) Let $$\phi(x,y,y')$$ denote an exact function (first integral) for the above ODE. Show that for this to be exact, it must have the following form $$F(x,y,y',y)=f(x,y,y')y+g(x,y,y')=0$$,  (1.3) where $$f$$ and $$g$$ are two functions of $$ (x,y,y')$$. Express $$f,g$$ in terms of $$\phi$$. The above result is the first condition for exactness for nonlinear 2nd-order ODEs. 10 points for the total derivative of $$\phi$$: $$\frac{d\phi}{dx}=\phi_x+p\phi_y +y''\phi_p$$.

10 points for partitioning the derivative as$$f=\phi_p$$ and $$ g=\phi_x +p\phi_y$$.  Question 2 

Next, the second condition for exactness of nonlinear 2nd-order ODEs is given as follows. Let $$p:= y'$$, we have $$f_{xx}+2pf_{xy}+p^2f_{yy}=g_{xp}+pg_{yp}-g_y$$ (1.4) $$f_{xp}+pf_{yp}+2f_y=g_{pp} $$ (1.5)

Derive the above exactness condition by eliminating $$\phi$$. To this end, differentiate the results in the previous question with respect to $$x,y,$$ and $$ p$$ to eliminate $$\phi$$. 15 points for equation 1.4(5 points for each partial, $$\phi_{xp},\phi_{yp},\phi_{xy}$$) 5 points for equation 1.5  Question 3 

Verify wheter (1.1) satisfy (a) The first condition for exactness.10 points (b) The second condition for exactness.10 points  Question 4 

Find a first integral $$\phi$$ whether or not (1.1) is exact. Hint: If (1.1) is not exact, use the integrating factor method with the integrating factor $$ h(x,y)=x^a y^b$$, (1.6)

5 points for constructing the $$f,g$$ equations. 5 points for $$a=3/2$$. 5 points for $$b=0$$. 5 points for solving $$\phi=x^2p+x^4y$$.  Question 5 

Find the solution $$y(x)$$ for (1.1)

5 points partial credit were awarded for using any solution technique (e.g., trial solution), particularly in cases where the $$\phi$$ derived in question 4 was incorrect and/or cumbersome.

10 points for solving for $$y(x)$$

10 points for any solution containing $$y=Ce^{-\frac{1}{3}x^3}$$. Some students solved $$\phi=x^2p+x^4y=0$$(separable) and some solved $$\phi=x^2p+x^4y=C$$ (more complicated). Both were accepted. --Egm6321.f09.TA 01:34, 10 November 2009 (UTC)