InterviewSolution
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InterviewSolution
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दर्शाइए कि अवकल समीकरण `(dy)/(dx)+(y^(2)+y+1)/(x^(2)+x+1)=0` का व्यापक हल `(x+y+1)=A(1-x-y-2xy)` है, जिसमे A एक प्राचल है। |
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Answer» दिया गया अवकल समीकरण `(dy)/(dx)+(y^(2)+y+1)/(x^(2)+x+1)=0` `implies (dy)/(y^(2)+y+1)+(dx)/(x^(2)+x+1)=0` समाकलन करने पर, `int(dy)/(y^(2)+y+1)+int(dx)/(x^(2)+x+1)=C` `implies int(dy)/(y^(2)+y+1+((1)/(2))^(2)-((1)/(2))^(2)) +int(dx)/(x^(2)+x+1+((1)/(2))^(2)-((1)/(2))^(2))=C` `implies int(dy)/((y+(1)/(2))^(2)+(1-(1)/(4)))+int(dx)/((x+(1)/(2))^(2)+(1-(1)/(4)))=C` `implies int(dy)/((y+(1)/(2))^(2)+(sqrt(3)/(2))^(2))+int(dx)/((x+(1)/(2))^(2)+(sqrt(3)/(2))^(2))=C` `implies (2)/(sqrt(3))tan^(-1)((y+(1)/(2))/(sqrt(3)/(2)))+(2)/(sqrt(3))tan^(-1)((x+(1)/(2))/(sqrt(3)/(2)))=C` `implies tan^(-1)((2y+1)/(sqrt(3)))+tan^(-1)((2x+1)/(sqrt(3)))=(sqrt(3)C)/(2)=K` (माना) `implies tan^(-1)[((2y+1)/(sqrt(3))+(2x+1)/(sqrt(3)))/(1-((2y+1)/(sqrt(3)))((2x+1)/(sqrt(3))))]=k` `impliestan^(-1)[((2y+1+2x+1)/(sqrt(3)))/(1-((4xy+2x+2y+1)/(3)))]=k` `implies (2sqrt(3)(x+y+1))/(3-(4xy+2x+2y+1))=tank` `implies (2sqrt(3)(x+y+1))/(2(1-x-y-2xy))=tank` `implies x+y+1=(1)/(sqrt(3))tank(1-x-y-2xy)` मान लीजिए `A=(1)/(sqrt(3))tank` जोकि स्वेच्छ अचर है। `implies x+y+1=A(1-x-y-2xy)` |
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