Find the solution of the differential equation (ey-x) d y=(ex-ey) d x

Question

Find the solution of the differential equation (ey-x) d y=(ex-ey) d x (A) ey ex=e2 x-ex2+c (B) ey ex=ex eex-eex+c (C) ey eex=ex eex-eex+c (D) eey ex=ex eex-eex+c

Tardigrade Admin · Accepted Answer

(ey-x) d y=(ex-ey) d x ⇒ ey ⋅ (d y/d x)=e2 x-ex ⋅ ey ⇒ ey ⋅ (d y/d x)+ex ⋅ ey=e2 x Let ey=z ⇒ ey ⋅ (d y/d x)=(d z/d x) ⇒ (d z/d x)+ex ⋅ z=e2 x IF =e∫ ezx ⋅ d x=eex ∴ Solution is Z ⋅ eex=∫ e2 x ⋅ eex ⋅ d x+ c Let ex=u ex ⋅ d x=d u ⇒ Z ⋅ eex=∫ u ⋅ eu d u+c=u eu-eu+c ⇒ ey ⋅ eex=eex(ex-1)+c

Q. Find the solution of the differential equation $(e^{y - x}) d y = (e^{x} - e^{y}) d x$

$e^{y} e^{x} = e^{2 x} - e^{x^{2}} + c$

$e^{y} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$

$e^{y} e^{e^{x}} = e^{x} e^{e^{x}} - e^{e^{x}} + c$

$e^{e^{y}} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$

Q. Find the solution of the differential equation (ey−x)dy=(ex−ey)dx

eyex=e2x−ex2+c

eyex=exeex−eex+c

eyeex=exeex−eex+c

eeyex=exeex−eex+c

(ey−x)dy=(ex−ey)dx ⇒ey⋅dxdy​=e2x−ex⋅ey ⇒ey⋅dxdy​+ex⋅ey=e2x Let ey=z⇒ey⋅dxdy​=dxdz​ ⇒dxdz​+ex⋅z=e2x IF =e∫ezx⋅dx=eex ∴ Solution is Z⋅eex=∫e2x⋅eex⋅dx+c Let ex=u ex⋅dx=du ⇒Z⋅eex=∫u⋅eudu+c=ueu−eu+c ⇒ey⋅eex=eex(ex−1)+c

Q. Find the solution of the differential equation $(e^{y - x}) d y = (e^{x} - e^{y}) d x$

$e^{y} e^{x} = e^{2 x} - e^{x^{2}} + c$

$e^{y} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$

$e^{y} e^{e^{x}} = e^{x} e^{e^{x}} - e^{e^{x}} + c$

$e^{e^{y}} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$