### Author Topic: TUT0202 Quiz 2  (Read 3626 times)

#### Jiwen Bi

• Jr. Member
• Posts: 7
• Karma: 3
##### TUT0202 Quiz 2
« on: October 04, 2019, 02:00:15 PM »
Jiwen Bi
$The\,required\,integrating\,factor\,is\, \mu =e^{3x} solve\,the\, differential\,eqution(3x^{2}+2xy+y^{3})e^{3x}dx+(x^{2}+y^{2})dy=0 \\ there\,exist\,a\,function\,\psi(x,y)such\,that\, \psi_{x}(x,y)=M(x,y)and\,\psi_{y}(x,y)=N(x,y)\\ this\, implies \,\psi_{x}(x,y)=(3x^{2}+2xy+y^{3})e^{3x}an\,\psi_{y}(x,y)=(x^{2}+y^{2})e^{3x}\\ intefrate\,the\,first\,of\,above\,eqution\\ \psi_{x}(x,y)=(3x^{2}+2xy+y^{3})e^{3x}\\ \psi(x,y)=\int(3x^{2}+2xy+y^{3})e^{3x}dx\\ =3y\int x^{2}e^{3x}dx+2y\int xe^{3x}dx+e^{3x}dx\\ =3y(\frac{x^{2}e^{3x}}{3}-\frac{2xe^{3x}}{9}+\frac{2e^{3x}}{27})+\frac{2y}{3}(xe^{3x}-\frac{e^{3x}}{3})+\frac{e^{3x}}{3}y^{3}+h(y) =x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}+h(y)\\ differentiate \,above eqution\, with \,respect \,to\, x \,and \,equate\,to\,N\\ \psi(x,y)=x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}+h(y)\\ \psi_{y}(x,y)=x^{2}e^{3x}+y^{2}e^{3x}+{h}'(y)\\ set \,\psi_{y}=N\,as\,follows:\\ x^{2}e^{3x}+y^{2}e^{ex}+{h}'(y)=x^{2}e^{3x}+y^{2}e^{3x}\\ {h}'(y)=0\\ this\,implies\,h(y)=0\\ substitude\,h(y)=0\,in\,equation\,\psi(x,y)=x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}+h(y).\\ \psi(x,y)=x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}+0\\ =x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}\\ then\,the\,solution\,of\,differential\,eqution\,is\\ x^{2}ye^{3x}+\frac{y^{3}}{3}e^{3x}=k\\ 3x^{2}ye^{3x}+y^{3}e^{3x}=3k\\ (3x^{2}y+y^{3})e^{3x}=C {3k = C}\\ Hence,\,the\,required\,solution\,of\,the\,differential\,eqution\,is(3x^{2}y+y^{3})e^{3x}=C$
« Last Edit: October 11, 2019, 01:34:20 PM by Jiwen Bi »