User:Egm6321.f09.Team6.dianafoster/HW1

Problem 6:
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.

a) First, it is known that: $$ M=\Phi_{x}=\frac{\partial \Phi(x,y)}{\partial x}$$ and  $$ N=\Phi_{y}=\frac{\partial \Phi(x,y)}{\partial y}$$

Taking the partial derivative for $$ \Phi_{x}$$ and $$ \Phi_{y}$$ for the following $$ \Phi(x,y)=6x^{4}+2y^{3/2}$$, yields: $$ M=\Phi_{x}=24x^3$$ and $$ N=\Phi_{y}=3y^{1/2}$$

Putting into the form $$ M+Ny'=0$$ so that the nonlinear 1st order ODE is in the desired form yields: $$ 24x^3+3y^{1/2}y'=0$$ b) Let $$ \Phi(x,y)=\frac{1}{3}x^{5}-4y^{3}$$ Taking the partial derivative for $$ \Phi_{x}$$ and $$ \Phi_{y}$$ yields:  $$ M=\Phi_{x}=\frac{5}{3}x^4$$ and  $$ N=\Phi_{y}=-12y^{2}$$ Putting into the form  $$ M+Ny'=0$$ yields:  $$ \frac{5}{3}x^4-12y^{2}y'=0$$ c) Let $$ \Phi(x,y)=x^6y^7+\frac{2}{5}y^{4}$$ Taking the partial derivative for $$ \Phi_{x}$$ and $$ \Phi_{y}$$ yields: $$ M=\Phi_{x}=6x^5y^7$$ and $$ N=\Phi_{y}=7x^6y^6+\frac{8}{5}y^3$$ Putting into the form $$ M+Ny'=0$$ yields: $$ 6x^5y^7+(7x^6y^6+\frac{8}{5}y^3)y'=0$$ d) Let $$ \Phi(x,y)=\frac{1}{2}xy^{3/2}+x^2y^5$$ Taking the partial derivative for $$ \Phi_{x}$$ and $$ \Phi_{y}$$ yields:  $$ M=\Phi_{x}=\frac{1}{2}y^{3/2}+2xy^5$$ and  $$ N=\Phi_{y}=\frac{3}{4}xy^{1/2}+5x^2y^4$$ Putting into the form  $$ M+Ny'=0$$ yields:  $$ (\frac{1}{2}y^{3/2}+2xy^5)+(\frac{3}{4}xy^{1/2}+5x^2y^4)y'=0$$