"(3ydx - 2xdy) + x^2y^{-1}(10ydx - 6xdy)=0"
"y(3ydx - 2xdy) + x^2(10ydx - 6xdy)=0"
"(3y^2+10x^2y)dx-(2xy+6x^3)dy=0"
"y(3y+10x^2)dx-2x(y+3x^2)dy=0"
"y(3y+10x^2)(2xdx)-4x^2(y+3x^2)dy=0"
"t=x^2, dt=2xdx"
"y(3y+10t)dt-4t(y+3t)dy=0"
"\\frac{dy}{dt}=\\frac{y(3y+10t)}{4t(y+3t)}"
This is an ODE of the homogeneous kind.
"y=tu, \\frac{dy}{dt}=u+t\\frac{du}{dt}"
Then:
"u+t\\frac{du}{dt}=\\frac{tu(3tu+10t)}{4t(tu+3t)}=\\frac{u(3u+10)}{4(u+3)}"
"t\\frac{du}{dt}=\\frac{u(3u+10)}{4(u+3)}-u=\\frac{3u^2+10u-4u^2-12}{4(u+3)}=\\frac{-u^2+10u-12}{4(u+3)}"
"-\\int\\frac{4(u+3)}{u^2-10u+12}du=\\int\\frac{dt}{t}"
"-\\frac{2}{13}((13+8\\sqrt{13})ln(-u+\\sqrt{13}+5)+(13-8\\sqrt{13})ln(u+\\sqrt{13}-5))=ln|u|+c"
"u=y\/x^2"
Answer:
"-\\frac{2}{13}((13+8\\sqrt{13})ln(-y\/x^2+\\sqrt{13}+5)+(13-8\\sqrt{13})ln(y\/x^2+\\sqrt{13}-5))="
"=ln|y\/x^2|+c"
Comments
Leave a comment