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Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/x dy=0,`given that `y=1`when `x=0` |
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Answer» `e^xdx(sqrt(1-y^2))+ydy(1/x)=0` `xe^xdxsqrt(1-y^2)+ydy=0` `xe^xdx=((-y)dy)/sqrt(1-y^2)` integrating both side `intxe^xdx=int(-y)dy/sqrt(1-y^2)` `intxe^xdx=x inte^xdx-intdx/dx*inte^xdxdx` `xe^x-inte^xdx=xe^x-e^x` `1/2t^(1/2)/(1/2)=sqrtt` `sqrt(1-y^2)` `xe^x-e^x=sqrt(1-y^2)+c` `c=-1` `xe^x-e^x=sqert(1-y^2)-1` `(x-1)e^x+1=sqrt(1-y^2)`. |
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