Question #308119

The differential equation



dy/dx= cos(x)(y^2 +6y+8)/(6y+16)




has an implicit general solution of the form F(x,y)=K



In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form


F(x,y)=G(x)+H(y)=K


Find such a solution and then give the related functions requested.


F(x,y)=G(x)+H(y)=

1
Expert's answer
2022-03-09T14:33:09-0500

The given equation can be separated as, cosx dx=6y+16(y+2)(y+4)dy\cos x~ dx = \dfrac{6y+16}{(y+2)(y+4)}dy.


Now using partial fractions for the term on the right hand side,


6y+16(y+2)(y+4)=Ay+2+By+46y+16=A(y+4)+B(y+2)\begin{aligned} \frac{6y+16}{(y+2)(y+4)} & = \frac{A}{y+2} + \frac{B}{y+4}\\ 6y+16 & =A(y+4) + B(y + 2)\\ \end{aligned}


Taking y=4y = -4 gives, B=4B = 4 and taking y=2y = -2 gives, A=2A = 2.

Therefore,


cosx dx={2y+2+4y+4}dy\cos x~ dx = \left\{\dfrac{2}{y+2} + \dfrac{4}{y+4}\right\}dy


Integrating both sides, we get

sinx=2log(y+2)+4log(y+4)sinx=log(y+2)2+log(y+4)4sinx+c=log{(y+2)2(y+4)4}e(sinx)+c=(y+2)2(y+4)4(y+2)2(y+4)4esinx=c1    (where c1=ec)\begin{aligned} \sin x &= 2\log (y+2) + 4\log (y+4) \\ \sin x &= \log (y+2)^2 + \log (y+4)^4\\ \sin x + c &= \log \{(y+2)^2 (y+4)^4\}\\ e^{(\sin x) + c} &= (y+2)^2(y+4)^4\\ (y+2)^2(y+4)^4 - e^{\sin x} &= c_1~~~~(\text{where }c_1 = e^c) \end{aligned}


Hence, F(x,y)=G(x)+H(y)=esinx+(y+2)2(y+4)4F(x,y)=G(x)+H(y)= - e^{\sin x} +(y+2)^2(y+4)^4 where


G(x)=esinx;H(y)=(y+2)2(y+4)4G(x)= - e^{\sin x}; H(y) = (y+2)^2(y+4)^4.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS