æ±å¾®åæ¹ç¨çé解
(4). ylnxdx-xlnydy=0
解ï¼å离åéå¾ï¼[(lnx)/x]dx-[(lny)/y]dy=0ï¼ å³æ(lnx)d(lnx)-(lny)d(lny)=0;
积åä¹ï¼å¾é解为ï¼ln²x-ln²y=c;
(5). dy/dx=6x²/(2y-siny)
解; å离åéå¾; (2y-siny)dy=6x²dx;
积åä¹å³å¾é解为ï¼y²+cosy=2x³+cï¼
(6). (x+2xy)dx+(1+x²)dy=0
è§£ï¼ x(1+2y)dx+(1+x²)dy=0
å离åéå¾ï¼[x/(1+x²)]dx+[1/(1+2y)]dy=0
å积åå¾ï¼ï¼1/2)â«d(1+x²)/(1+x²)+(1/2)â«d(1+2y)/(1+2y)=0
积åä¹å¾é解为ï¼ln(1+x²)+ln(1+2y)=c₁
æåæ: (1+x²)(1+2y)=cï¼å
¶ä¸c=e^c₁ï¼
(7). è¥â(1-x²)æ¯xf(x)çä¸ä¸ªåå½æ°ï¼åâ«ã0,1ã[1/f(x)]dx=?
解ï¼xf(x)=[â(1-x²)]'=-x/â(1-x²)ï¼â´f(x)=-1/â(1-x²)ï¼
â´â«ã0,1ã[1/f(x)]dx=-â«ã0,1ãâ(1-x²)dx=-[(x/2)â(1-x²)+(1/2)arcsinx]ã0,1ã=-Ï/4;
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