东北师大《常微分方程》习题解答.pdf

1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 课后答案网 w w w .k h d a w .c o m 2 ? 1.2 1 ? (1) xdxydy= ? 1 22 2 1 2 1 cxy+= ? cyx= 22 (2) yy dx dy ln= ? 1, 0=yy?1, 0yy?dx yy dy = ln ? ?0ln,lnln 1 1 =+=cceeeycxy xxc ? x ce ey = (3) yx e dx dy = ? ? dxedye xy =?cee xy = (4) 0cottan=xdyydx ? x y dx dy cot tan =?0=y?0y?dx x x dy y y cos sin sin cos =. ? 11 cossinln,coslnsinlncxycxy=+=, ?0,cossin 1 =ccexy c 2? (1) 1)0(),1(=yyy dx dy ? 1, 0=yy?1, 0yy?dxdy yy = ) 1 1 1 (? ? 0, 1 , 1 ln 1 1 = += cceee y y cx y y xxc ?1)0(=y? 0=c?1=y? (2) 1)0(, 02) 1( 22 =+yxyyx ? 0, 1 2 2 2 = =y x xy dx dy ?0y?dx x x y dy 1 2 22 =? 课后答案网 w w w .k h d a w .c o m 3 ? cx y +=1ln 1 2 ?1)0(=y? 1=c? 11ln 1 2 + = x y? (3) 0)2(,33 2 =yyy ? 0=y?0y?dx y dy = 3 2 3 ? ? 3 3 1 )(,cxycxy+=+= ?0)2(=y? 2=c? 3 )2( = xy?0=y? (4) 1) 1 (, 0)()( 2222 =+ydyyxxdxxyy ? 0, 0=yx?0, 0yx?0 11 22 = + + dy y y dx x x ? ? 0,ln 1 ln 1 1111 1 1 =+ cceee y x cy y x x yxyxc ?1) 1 (=y? 2 =ec? yx ee y x 11 2 =? 4? 011 22 =+dyxydxyx ?) 11(1),11(1=xyyx? ?1, 1yx?0 11 22 = + dy y y dx x x ? )0(11 22 =+ccyx 6? x ? ?(? x ?)?(1,2). ? )(xyy=?),(yx? )( xXyyY=? x ? y y xa=? 课后答案网 w w w .k h d a w .c o m 4 0 =xx y y x?2) 1 (,=y x y y ?0,=cexy c ? c e=2? c = ln2 , ? ?2=xy 7? ?1? ? 4 ?2 ? 12 ? ?2? ? 3 ? 4 10? 5 ? 4 104? ? ? t ? q (t) , ?0=kkq dt dq ? kt ceq = ? t = 0 ? 0 q,? kt eqq 0 = ?1? ? 0 2)4(qq= ? 0 4 0 2qeq k = ? 4 2ln =k 0 4 2ln 12 0 12 0 8)12(qeqeqq k = ?2? 45 0 43 0 104)5(,10)3(= kk eqqeqq ? 2 4ln =k? ? 4 0 2 4ln 3 0 3 0 108)3(=qeqeqq k ? 3 0 1025. 1=q ? 1.3 1 ? (2) 0)2( 22 =+dyxdxxyy ? 2 )()(2 x y x y dx dy = ? x y u =? 2 2uu dx du xu=+ ? ) 1 , 0() 1 11 (= u x dx du uu ? xc u u 1 ln 1 ln= ? 1 1 1 = xc xc u ?0,)(=ycyxyx? 课后答案网 w w w .k h d a w .c o m 5 (4) x y xyyxtan= ? x y x y dx dy tan= ? x y u =? u u u dx du x cos sin tan= ?)0(sincot=u x dx udu ? cxu=sin ?cx x y =sin (5) x yx yxyyx + +=ln)( ? x yx x y x y dx dy+ +=ln)1 ( ? x y u =? )1ln()1 (uu dx du x+= ?1, 0uu? x dx uu du = +)1ln()1 ( ? cxu=+)1ln( ?cx x y =+)1ln( (6) yyxyx+= 22 ? x y x y dx dy += 2 )(1 ? x y u =? 2 1u dx du x= ? ) 11( 1 2 ?, ?|( )|f x0? , ?(3)? ; ?y0? , ?(6)? ; ?x = CxCx Cx y ,)( , 0 2 3 ? ? ? ? ? = .,)( , 0 2 3 CxCx Cx y 4. ? 2 yx dx dy =?0)0(=y ?: )(),(),(),( 3210 xxxx ? ?0)0()( 0 =yx 2 0 1 2 1 )0(0)(xdssx x ? =+= 52 0 22 2 20 1 2 1 ) 2 1 (0)(xxdsssx x =+= ? . 4400 1 160 1 20 1 2 1 ) 20 1 2 1 (0)( 11852 0 252 3 xxxxdssssx x +=+= ? 5. ? 22 xy dx dy =?1)0(=y ?: )(),(),( 210 xxx ? ?1)0()( 0 =yx 3 0 1 3 1 1)1 (0)(xxdssx x +=+= ? . 63 1 15 2 6 1 1) 3 1 1 1)( 7542 0 223 2 xxxxxdssssx x +=+= ? 6. ?2 . 2?n?)(x n ?)(x? ? (9) 课后答案网 w w w .k h d a w .c o m 17 ? : 1 0 )!1( )()( + + n n n xx n MN xx ? ? ? ? += x x dsssfyx 0 )(,()( 0 ? ? 00 )(yx=, ?, 2, 1)(,()( 0 10 =+= ? ndsssfyx x x nn ? 00 0 )(,()()(xxMdsssfxx x x ? ? 1 0 )!1( )()( + + n n n xx n MN xx ? 2 0 1 1 0 1 1 )!2( )!1( )(,()(,()()( 0 0 + + + + + + + ? ? n n x x n n z x nn xx n MN dsxs n MN dsssfssfxx ?,?n?, ? 1 0 )!1( )()( + + n n n xx n MN xx ? . 7. ?, ? : (1) 4 ?)( 3 x ? 2 1 =x?1=x?; (2) 5 ?)( 2 x? 4 1 =x?. ? ? (1) ? 2 yx dx dy =, 0)0(=y ?1, 1:yxR ? ?, ?,? 0 hx ?),min( 0 M b ah =, = 2 ),(max yxM Ryx 2. ?, 1, 1=ba ? 2 1 0 =h,? 课后答案网 w w w .k h d a w .c o m 18 ? 2 1 x. ? 1 0 )!1( )()( + + n n n xx n MN xyxy ?N?. ? , 22= y y f ? , 2=N ? 2 1 =x? , ? 24 1 ) 2 1 ( ! 4 22 )()( 4 3 3 = xyxy. ?1=x? , ? 3 2 ) 1 ( ! 4 22 )()( 4 3 3 = xyxy. (2) ? 22 xy dx dy =,1)0(=y ?11, 1:yxR ? ?, ?, ?: 0 hx ?),min( 0 M b ah=, = 22 ),(max xyM Ryx 4. ?, 1, 1=ba ? 4 1 0 =h,? ? 4 1 x. ? 1 0 )!1( )()( + + n n n xx n MN xyxy ?N?. ? , 22= y y f ?, 2=N ? 24 1 ) 4 1 ( ! 3 24 )()( 3 2 2 = xyxy. 8. ?bxa, +y ? , ? (2.1) ?, ?)(),( 21 xyxy, ?)()( 0201 xyxy, ? )()( 21 xyxy, bxx 0 . 课后答案网 w w w .k h d a w .c o m 19 ? ? ? =)