User:Egm6321.f11.team2.Xia/RP3.3

= R*3.3 -First Integral =

Given
Consider a class of N1-ODEs of the form in (1)p.13-2: $$\displaystyle \overline{b}(x,y)c(y)y'+a(x)\overline{c}(x,y)=0$$ (3.3.1)

and $$\displaystyle a(x)=\sin x^{3}$$ $$\displaystyle b(x)=\cos x$$ $$\displaystyle c(y)=\exp(2y)$$ ---

Find
1. Find an N1-ODE of the form (1) that is either exact of can be made exact by IFM. 2. Find the first integral $$ \phi(x,y)=k$$

Solution
Since $$\displaystyle \overline{b}(x,y)=\int^{x}b(s)ds+k_{1}(y)$$ $$\displaystyle \overline{c}(x,y)=\int^{y}c(s)ds+k_{2}(x)$$ Substitute the given equations in the above two equations, we have $$\displaystyle \overline{b}(x,y)=\int^{x}\cos sds+k_{1}(y)=\sin x+d_{1}$$ $$\displaystyle \overline{c}(x,y)=\int^{y}\exp(2s)ds+k_{2}(x)=\frac{1}{2}\exp(2y)+d_{2}$$ (3.3.2) where $$ \displaystyle d_1, d_2$$ are integral constant. Then Let $$\displaystyle d_1=d_2=0$$ and substitute the above equations to (3.3.1) $$\displaystyle \sin x\exp(2y)y'+\frac{1}{2}\exp(2y)\sin x^{3}=0$$ hence $$\displaystyle M(x,y)= \frac{1}{2}\exp(2y)\sin x^{3}$$ (3.3.3) $$\displaystyle N(x,y)= \sin x\exp(2y)$$ (3.3.4) It is easy to know that $$\displaystyle M_y (x,y) \neq N_x (x,y)$$ Recall IFM $$\displaystyle h(x)=\frac{M_{y}(x,y)-N_{x}(x,y)}{N(x,y)}=\frac{\exp(2y)(\sin x^{3}-\cos x)}{\sin x\exp(2y)}=\frac{\sin x^{3}-\cos x}{\sin x}$$ (3.3.5) So, it can be made exact by IFM.

Since $$\displaystyle \exp(\int h(x)dx)=\exp(\frac{x}{2}-\frac{\sin2x}{4}-\ln(\sin x))$$ we have $$\displaystyle \phi_{x}=M(x,y)\exp(\int h(x)dx)=\frac{1}{2}\exp(\frac{x}{2}-\frac{\sin2x}{4}+2y)\sin x$$ $$\displaystyle \phi_{y}=N(x,y)\exp(\int h(x)dx)=\exp(\frac{x}{2}-\frac{\sin2x}{4}+2y)$$ then $$\displaystyle \phi=\frac{1}{2}\exp(\frac{x}{2}-\frac{\sin2x}{4}+2y)=k$$ (3.3.6)