Vrai / Faux sur les équations différentielles

Vrai / Faux — Équations Différentielles Série 1

  1. Q11. Soit \((E): y' + 3y = 6\). La solution \(f\) de \((E)\) telle que \(f(0) = 5\) est définie par \(f(x) = 3e^{-3x} + 2\).

    • Vrai

    • Faux

    Remarque .10003LD{ue+EdCé=()3cA2^f/o h-l}sr,amyxitp0nv5g.050K040b0E0w0B0v0z0e0J0I0v0M0w0P0k0M0k0C0E0z0f0w0i0;050B0K0E0(0p0z0E0B0B0l010E0C0I0J0x0F0E0J0f0H0104090m0m0h0n090n050u0_0{040w0f0B0J0w0^0`0|0~10121416181a1c1e1g0G0w0l0w0j0f0s0d0y0o0H0A0w0g0w0r1l1n1p0M040Q0w0q0N0f0p1w1$0w0}0 11131517191b1d1f0m0t0m0L0n1L0w0O1!1o1y040D0w0%0w0J0C0v0e1,1/1y1;1A1@1D1`1G1}1g0j240o271$1(0w0c0%1.1x0{2m1?1C1_1F1|1I1 0m0H231M0o1P1R1T1V1X1Z1m281q1)0a1n0S.

  2. Q12. Soit \(f\) une solution de \(y' = -2y + 8\).

    Affirmation : La limite de \(f\) en \(+\infty\) dépend de la condition initiale \(f(0)\).

    • Vrai

    • Faux

    Remarque .L{ue+dC4é=()c2^f/o h-l}srq,amQyxitp0nvg.050I040a0B0s0x0r0v0c0H0G0r0K0s0M0i0K0i0y0B0v0d0s0d0x0H0s050x0I0B0#0m0v0B0x0x0j010B0y0G0H0t0C0B0H0d0F0104090k0p0k0F0l0s0j0s0g0d0o0b0u0n0F0w0s0e0s0h090l050q0^0`040N0s0D0c0`0f0?1E0{0}0 11131517191b1d1f0F0s090H0r1w090G0K0p0H0E1A1C1O040A1N0_1P0~10121416181a1c1e1g1i1k1$1(1z1B1D1_041N0H0y0!0M040z0c0d0v0-0s2k0.0U151;2f2h2d0}0s0L0,0d0c0y0s0f0.0@1_0s0|1{1S1~1V211Y0k0g1:2b1F0s0k0f0!0m2F2H1O2K1Q1|1T1 1W221f1h0J0l2U1=0l0N1C0P.

  3. Q13. On considère l'équation \(y' = ay\). Si \(f(0) = 1\) et \(f(1) = e^2\), alors \(a = 2\).

    • Vrai

    • Faux

    Remarque .10003L{uebé(=)cA2^f/o h}lsr,amxitp0n1v.050D040b0y0r0v0q0u0d0C0B0q0F0r0e0v0C0r050v0D0y0W0k0u0y0v0v0i010y0w0B0C0s0z0y0C0e0A0104090h0o0h0A0j0r0i0X0n0c0y0A0t090j050p0%0)040I0r0l0H0e0k0#1j0*0,0.0:0=0@0_0{0}0 11130E16180G1f1h1t040r0h0H0g0?0o0B0g160e0!0$0(1u0-0/0;0?0^0`0|0~10120h0G1H190N180e0n0m1K1i1!040x0r0V210f0T0e0F1Y1t0r0+1$1x1)1A1,1D0h1@0r1{1g1}1k1m0a1h0K.

  4. Q14. Soit \(f\) la solution de \(y' = y - 1\) passant par le point \(A(0;2)\).

    Affirmation : La tangente à la courbe de \(f\) au point \(A\) a pour équation \(y = x + 2\).

    • Vrai

    • Faux

    Remarque .10003Lue+dé(=)cA2^f/o hOlsrq,amyxitp0n1g.050F040b0z0r0v0q0u0c0E0D0q0H0r0d0v0E0r050v0F0z0Y0k0u0z0v0v0i010z0w0D0E0s0A0z0E0d0C0104090h0o0h0C0j0r0i0Z0n0C0r0e0r0I090j050p0)0+040K0r0t0Y0P0(0*0,0.0:0=0@0_0{0}0 1113150G181a0m1i1k1m0H040Z0$1u0+0r0-0/0;0?0^0`0|0~10121400161J1b0C1M1l1v040y0r0f0X0k0%1O1V1x1Y1A1#1D1(1G1+1I191g1:1O1o0r0b000g0x0c0}0W0Y0f0d0r0.0r0E0+0J0d0H0~0Z0Y1T1w1X1z1!1C1%1F0h0l2a1=0Z0#1|1v1~2B1Z1B1$1E1)0B280I0r090k1_0$1d1f1L1j1;1n1p0a1k0M.

  5. Q15. Toute solution de l'équation différentielle \(y' = -y + e^x\) est strictement croissante sur \(\mathbb{R}\).

    • Vrai

    • Faux

    Remarque .LD{ueR+Cbé=()c2^f/o h-l}sr,amP;yxitp0n1g.050J040a0B0t0y0s0w0d0I0H0s0L0t0N0j0L0j0z0B0w0e0t0e0y0I0t050y0J0B0$0n0w0B0y0y0k010B0z0H0I0u0C0B0I0e0G0104090l0F0t0k0t0h0e0p0c0v0G0x0t0g0t090q0,0n0c0M0x0c0o0x1n0G090m050r0_0{040O0t0D0s0d0z0@1M0|0~10121416181a1c1e1g0h1j0t0M1I1K1W040A1V0`1X0 11131517191b1d1f1h001-0v1n1p1r1t1v1x0B1z1B1D1F0p1H1J1L1_040:0?0^1_0t0}1{1!1~1%211*240l0K0m260M2b1w1y1A1C1E262I2e2K2h0t020w0I0E0t0K1:2m1N1P0b0#0n1^0{2t1Y1|1#1 1(221g0F2Z1=0t0L000;0?0`0y2t0z0s0H100{1b0U1T2*1`1Z1}1$201)2309200u0i0i0c0f0x2^2n0O1K0Q.

  6. Q16. L'équation \(y' = 0,1y(20 - y)\) admet deux solutions constantes : les fonctions \(y = 0\) et \(y = 20\).

    • Vrai

    • Faux

    Remarque .10003LeRud)é=(c2f/o h-lsr,amywxitp0n1vg.050D040b0c0t0p0t0o0s0e0C0B0o0F0P0k0X0t0C0w0F0C0O0p0H0h0u0B0m0B0c0)0p050t0D0(0p0k0s0w0t0t0i010w0/0C0q0x0w0*0A0104090j0y000p0i0p0E090g050n0`0(040v0p0f0X0k0^1r0F0}0 1113150B17191b1d1f0E0v0G0y0j0l0E0p0r0p0y0g1j1l0p090d0B0I0q0%0u0u0o0z1W1Z1m1o1q0{0F040p0o0e1z1^1C10121416181a0c1c1e1g1Z1S1n1p1A040J0p0a1p0L.

  7. Q17. La fonction \(f(x) = e^{-x} + x - 1\) est solution de l'équation \(y' + y = x\).

    • Vrai

    • Faux

    Remarque .10003D{e+(=)c^f/- hOl}osrmaxitpn1.050A040p0B0n0w0n050t0A0w0I0i0q0w0t0t0g010w0u0y0z0o0v0w0z0d0x0104090f0k000f0x0h0n0g0n0m0d0j0c0m0x0r0n0e0n0C090h050l0N0P040D0n0b0s0B0i0L1a0Q0S0U0W0Y0!0$0(0*0,0.0:0=0@130k1x0^0n0f0{0}0 11130C1y1D0|0~10120n0x0`140@0_0x16181k1c0n0a180F.

  8. Q18. Soit \(f\) une solution de \(y' = ay + b\) avec \(a < 0\). On note \(y_0\) la solution constante.

    Affirmation : Si \(f(0) > y_0\), alors la fonction \(f\) est décroissante et convexe sur \(\mathbb{R}\).

    • Vrai

    • Faux

    Remarque .10003LD{uedCb=()éc2^f/o h-l}Os_r,am;yxitp0nvg.050K040b0D0t0z0s0w0e0J0I0s0M0t0n0$0z0J0D0M0J0f0t0f0+0t050z0K0-0(0w0D0z0z0j010D0B0I0J0u0E0D0/0H0104090k0G0A0L0t0j0t0v090q0B0D0n0d0i0x0d0D0x090l050r0_0-040P0t0T0V0X0Z0#0%0O0m0M0m1p0w0:0=0J0@1C0%0n0}0 11131517190f1b1d0k0q0k0H0l1j0t0h0f0p1v0H0x1l0t1n1p1r1t1v1x1z1B0`0M040t0D0N0f0n1W280|0~10121416181a1c1e0h1?1/0L1=0v0t1g1i020O0J0F0t0L1y1A1X1E0t0y0%0U0^2h1Z2j1$2m1)2p1-0q001:1=1k0D1^1`0D1|0t020w2D2F2H271D0t0k0n132g0{2R1#2l1(2o1+2q0k0U2-2/2G262J0;1V2P2|1!2k1%2n1*1,2r2,2C2E382I28040l0C0t0g0$2f3d1Y3f2T303j330q2;3a1U3v0m0n0B0s0I0 0-0/1F0c0:0K0Y0z3u3z2i2~3h2V322X002Z1;1?0D0p0o2(1{1}2B373H3r2@2_0B2{3A2S2 3i2W1e3/0o3m3_393{0f3c1X3#3g2U313k0k2s3^3o3`1D3t3v3x3 4f3C433)1e3G4a2?3J0)0M2d0H0f1F0a1A0R.

  9. Q19. On considère l'équation \((E): y' = 2y - e^{2x}\).

    Affirmation : Toutes les courbes des solutions de \((E)\) possèdent la même tangente au point d'abscisse \(0\).

    • Vrai

    • Faux

    Remarque .L{ue+EdC)é=(3c2^/o h}lsrq,amyxitp0n1g.050G040a0A0s0w0r0v0c0F0E0r0I0s0K0j0I0j0x0A0v0d0s0d0w0F0s050w0G0A0Z0n0v0A0w0w0k010A0x0E0F0t0B0A0F0d0D0104090l0C0s0k0s0l0h0s0e0s0D1l0s0J0i0d0p0b0o0D0u090i050q0?0^040L0s0f0Z0=0@0_0{0}0 11131517191b1d1o1h0H1z1B1D0I040z0;1$0s0`0|0~10121416181a1c1e0l0H0i1g0s1k1m0J1!1C1L040-0:1K0^1,1N1/1Q1=1T1^1d0C001{1}1h0o200s0m231$1F0s0a000j0y0c170X0Z0g0,0{0s0F0^0K0d0I180-0/1*1L2b1.1P1;1S1@1V1e1~0l2o1p0m0i1o1m1j1p1r2s251)2z0E1*0F0x0Y0K040g0j0G2L0g0s2E2Q2a1-1O1:1R1?1U1_0h2/1E1#2^2`260l0g0Y0n322F0Q0S0U0W0Y0i0L1B0N.

  10. Q20. Soit \(f\) une solution non nulle de \(y' = ay\).

    Affirmation : Pour tous réels \(x_1\) et \(x_2\), \(f(x_1 + x_2) = f(x_1) \times f(x_2)\) si et seulement si \(f(0) = 1\).

    • Vrai

    • Faux

    Remarque .10003L{ue+dC=()éc2^f/o h}lOs_rq,àamxitp0n1vg.050I040b0D0s0x0r0v0d0H0G0r0K0s0e0x0H0s050x0I0D0$0m0v0D0x0x0i010D0z0G0H0t0E0D0H0e0F0104090j0p0j0F0k0s0i0s0h0e0o0c0D0F0u090k050q0-0/040O0s0w0$0T0,0.0:0=0@0_0{0}0 11131517190F0y0L0s0f0s1N0n1c1e1g1i0D1a1O1Q1S0y1U1m1o1q1A040%0*1z0/0s0;0?0^0`0|0~10121416181!0L0k1M1(1V1f0o0n1h1j221$1T0k1*1p1r0K1t0s0b000l0N0D0v0}0l0+2j1?1C1_1F1|1I1 1L2d1R2f1d0s2523251U1n2i1-0s0l0A0d0G0M0D0Y0s0C2v1A2x1^1E1{1H1~1K0j0h2I0h292O1,1s0B0U0r0}2#1=1@1D1`1G1}1J202/1e0J2?2j1.0r0d2}1B2(312B2,352I0L391-1u0h0r0E0E0e3e2%302A2+341L3n1s0%0)0s0K0#3G0d0v0v0e2_1;3f3x2*332D2.3l3C2k2_0g0#0m3v2 2z3S2C2-190J273m1+3a1u0a1p0Q.

Vrai / Faux — Équations Différentielles Série 2

  1. Q11. Soit \((E): y' + 3y = 6\). La solution \(f\) de \((E)\) telle que \(f(0) = 5\) est définie par \(f(x) = 3e^{-3x} + 2\).

    • Vrai

    • Faux

    Remarque .10003LD{ue+EdCé=()3cA2^f/o h-l}sr,amyxitp0nv5g.050K040b0E0w0B0v0z0e0J0I0v0M0w0P0k0M0k0C0E0z0f0w0i0;050B0K0E0(0p0z0E0B0B0l010E0C0I0J0x0F0E0J0f0H0104090m0m0h0n090n050u0_0{040w0f0B0J0w0^0`0|0~10121416181a1c1e1g0G0w0l0w0j0f0s0d0y0o0H0A0w0g0w0r1l1n1p0M040Q0w0q0N0f0p1w1$0w0}0 11131517191b1d1f0m0t0m0L0n1L0w0O1!1o1y040D0w0%0w0J0C0v0e1,1/1y1;1A1@1D1`1G1}1g0j240o271$1(0w0c0%1.1x0{2m1?1C1_1F1|1I1 0m0H231M0o1P1R1T1V1X1Z1m281q1)0a1n0S.

  2. Q12. Soit \(f\) une solution de \(y' = -2y + 8\).

    Affirmation : La limite de \(f\) en \(+\infty\) dépend de la condition initiale \(f(0)\).

    • Vrai

    • Faux

    Remarque .L{ue+dC4é=()c2^f/o h-l}srq,amQyxitp0nvg.050I040a0B0s0x0r0v0c0H0G0r0K0s0M0i0K0i0y0B0v0d0s0d0x0H0s050x0I0B0#0m0v0B0x0x0j010B0y0G0H0t0C0B0H0d0F0104090k0p0k0F0l0s0j0s0g0d0o0b0u0n0F0w0s0e0s0h090l050q0^0`040N0s0D0c0`0f0?1E0{0}0 11131517191b1d1f0F0s090H0r1w090G0K0p0H0E1A1C1O040A1N0_1P0~10121416181a1c1e1g1i1k1$1(1z1B1D1_041N0H0y0!0M040z0c0d0v0-0s2k0.0U151;2f2h2d0}0s0L0,0d0c0y0s0f0.0@1_0s0|1{1S1~1V211Y0k0g1:2b1F0s0k0f0!0m2F2H1O2K1Q1|1T1 1W221f1h0J0l2U1=0l0N1C0P.

  3. Q13. On considère l'équation \(y' = ay\). Si \(f(0) = 1\) et \(f(1) = e^2\), alors \(a = 2\).

    • Vrai

    • Faux

    Remarque .10003L{uebé(=)cA2^f/o h}lsr,amxitp0n1v.050D040b0y0r0v0q0u0d0C0B0q0F0r0e0v0C0r050v0D0y0W0k0u0y0v0v0i010y0w0B0C0s0z0y0C0e0A0104090h0o0h0A0j0r0i0X0n0c0y0A0t090j050p0%0)040I0r0l0H0e0k0#1j0*0,0.0:0=0@0_0{0}0 11130E16180G1f1h1t040r0h0H0g0?0o0B0g160e0!0$0(1u0-0/0;0?0^0`0|0~10120h0G1H190N180e0n0m1K1i1!040x0r0V210f0T0e0F1Y1t0r0+1$1x1)1A1,1D0h1@0r1{1g1}1k1m0a1h0K.

  4. Q14. Soit \(f\) la solution de \(y' = y - 1\) passant par le point \(A(0;2)\).

    Affirmation : La tangente à la courbe de \(f\) au point \(A\) a pour équation \(y = x + 2\).

    • Vrai

    • Faux

    Remarque .10003Lue+dé(=)cA2^f/o hOlsrq,amyxitp0n1g.050F040b0z0r0v0q0u0c0E0D0q0H0r0d0v0E0r050v0F0z0Y0k0u0z0v0v0i010z0w0D0E0s0A0z0E0d0C0104090h0o0h0C0j0r0i0Z0n0C0r0e0r0I090j050p0)0+040K0r0t0Y0P0(0*0,0.0:0=0@0_0{0}0 1113150G181a0m1i1k1m0H040Z0$1u0+0r0-0/0;0?0^0`0|0~10121400161J1b0C1M1l1v040y0r0f0X0k0%1O1V1x1Y1A1#1D1(1G1+1I191g1:1O1o0r0b000g0x0c0}0W0Y0f0d0r0.0r0E0+0J0d0H0~0Z0Y1T1w1X1z1!1C1%1F0h0l2a1=0Z0#1|1v1~2B1Z1B1$1E1)0B280I0r090k1_0$1d1f1L1j1;1n1p0a1k0M.

  5. Q15. Toute solution de l'équation différentielle \(y' = -y + e^x\) est strictement croissante sur \(\mathbb{R}\).

    • Vrai

    • Faux

    Remarque .LD{ueR+Cbé=()c2^f/o h-l}sr,amP;yxitp0n1g.050J040a0B0t0y0s0w0d0I0H0s0L0t0N0j0L0j0z0B0w0e0t0e0y0I0t050y0J0B0$0n0w0B0y0y0k010B0z0H0I0u0C0B0I0e0G0104090l0F0t0k0t0h0e0p0c0v0G0x0t0g0t090q0,0n0c0M0x0c0o0x1n0G090m050r0_0{040O0t0D0s0d0z0@1M0|0~10121416181a1c1e1g0h1j0t0M1I1K1W040A1V0`1X0 11131517191b1d1f1h001-0v1n1p1r1t1v1x0B1z1B1D1F0p1H1J1L1_040:0?0^1_0t0}1{1!1~1%211*240l0K0m260M2b1w1y1A1C1E262I2e2K2h0t020w0I0E0t0K1:2m1N1P0b0#0n1^0{2t1Y1|1#1 1(221g0F2Z1=0t0L000;0?0`0y2t0z0s0H100{1b0U1T2*1`1Z1}1$201)2309200u0i0i0c0f0x2^2n0O1K0Q.

  6. Q16. L'équation \(y' = 0,1y(20 - y)\) admet deux solutions constantes : les fonctions \(y = 0\) et \(y = 20\).

    • Vrai

    • Faux

    Remarque .10003LeRud)é=(c2f/o h-lsr,amywxitp0n1vg.050D040b0c0t0p0t0o0s0e0C0B0o0F0P0k0X0t0C0w0F0C0O0p0H0h0u0B0m0B0c0)0p050t0D0(0p0k0s0w0t0t0i010w0/0C0q0x0w0*0A0104090j0y000p0i0p0E090g050n0`0(040v0p0f0X0k0^1r0F0}0 1113150B17191b1d1f0E0v0G0y0j0l0E0p0r0p0y0g1j1l0p090d0B0I0q0%0u0u0o0z1W1Z1m1o1q0{0F040p0o0e1z1^1C10121416181a0c1c1e1g1Z1S1n1p1A040J0p0a1p0L.

  7. Q17. La fonction \(f(x) = e^{-x} + x - 1\) est solution de l'équation \(y' + y = x\).

    • Vrai

    • Faux

    Remarque .10003D{e+(=)c^f/- hOl}osrmaxitpn1.050A040p0B0n0w0n050t0A0w0I0i0q0w0t0t0g010w0u0y0z0o0v0w0z0d0x0104090f0k000f0x0h0n0g0n0m0d0j0c0m0x0r0n0e0n0C090h050l0N0P040D0n0b0s0B0i0L1a0Q0S0U0W0Y0!0$0(0*0,0.0:0=0@130k1x0^0n0f0{0}0 11130C1y1D0|0~10120n0x0`140@0_0x16181k1c0n0a180F.

  8. Q18. Soit \(f\) une solution de \(y' = ay + b\) avec \(a < 0\). On note \(y_0\) la solution constante.

    Affirmation : Si \(f(0) > y_0\), alors la fonction \(f\) est décroissante et convexe sur \(\mathbb{R}\).

    • Vrai

    • Faux

    Remarque .10003LD{uedCb=()éc2^f/o h-l}Os_r,am;yxitp0nvg.050K040b0D0t0z0s0w0e0J0I0s0M0t0n0$0z0J0D0M0J0f0t0f0+0t050z0K0-0(0w0D0z0z0j010D0B0I0J0u0E0D0/0H0104090k0G0A0L0t0j0t0v090q0B0D0n0d0i0x0d0D0x090l050r0_0-040P0t0T0V0X0Z0#0%0O0m0M0m1p0w0:0=0J0@1C0%0n0}0 11131517190f1b1d0k0q0k0H0l1j0t0h0f0p1v0H0x1l0t1n1p1r1t1v1x1z1B0`0M040t0D0N0f0n1W280|0~10121416181a1c1e0h1?1/0L1=0v0t1g1i020O0J0F0t0L1y1A1X1E0t0y0%0U0^2h1Z2j1$2m1)2p1-0q001:1=1k0D1^1`0D1|0t020w2D2F2H271D0t0k0n132g0{2R1#2l1(2o1+2q0k0U2-2/2G262J0;1V2P2|1!2k1%2n1*1,2r2,2C2E382I28040l0C0t0g0$2f3d1Y3f2T303j330q2;3a1U3v0m0n0B0s0I0 0-0/1F0c0:0K0Y0z3u3z2i2~3h2V322X002Z1;1?0D0p0o2(1{1}2B373H3r2@2_0B2{3A2S2 3i2W1e3/0o3m3_393{0f3c1X3#3g2U313k0k2s3^3o3`1D3t3v3x3 4f3C433)1e3G4a2?3J0)0M2d0H0f1F0a1A0R.

  9. Q19. On considère l'équation \((E): y' = 2y - e^{2x}\).

    Affirmation : Toutes les courbes des solutions de \((E)\) possèdent la même tangente au point d'abscisse \(0\).

    • Vrai

    • Faux

    Remarque .L{ue+EdC)é=(3c2^/o h}lsrq,amyxitp0n1g.050G040a0A0s0w0r0v0c0F0E0r0I0s0K0j0I0j0x0A0v0d0s0d0w0F0s050w0G0A0Z0n0v0A0w0w0k010A0x0E0F0t0B0A0F0d0D0104090l0C0s0k0s0l0h0s0e0s0D1l0s0J0i0d0p0b0o0D0u090i050q0?0^040L0s0f0Z0=0@0_0{0}0 11131517191b1d1o1h0H1z1B1D0I040z0;1$0s0`0|0~10121416181a1c1e0l0H0i1g0s1k1m0J1!1C1L040-0:1K0^1,1N1/1Q1=1T1^1d0C001{1}1h0o200s0m231$1F0s0a000j0y0c170X0Z0g0,0{0s0F0^0K0d0I180-0/1*1L2b1.1P1;1S1@1V1e1~0l2o1p0m0i1o1m1j1p1r2s251)2z0E1*0F0x0Y0K040g0j0G2L0g0s2E2Q2a1-1O1:1R1?1U1_0h2/1E1#2^2`260l0g0Y0n322F0Q0S0U0W0Y0i0L1B0N.

  10. Q20. Soit \(f\) une solution non nulle de \(y' = ay\).

    Affirmation : Pour tous réels \(x_1\) et \(x_2\), \(f(x_1 + x_2) = f(x_1) \times f(x_2)\) si et seulement si \(f(0) = 1\).

    • Vrai

    • Faux

    Remarque .10003L{ue+dC=()éc2^f/o h}lOs_rq,àamxitp0n1vg.050I040b0D0s0x0r0v0d0H0G0r0K0s0e0x0H0s050x0I0D0$0m0v0D0x0x0i010D0z0G0H0t0E0D0H0e0F0104090j0p0j0F0k0s0i0s0h0e0o0c0D0F0u090k050q0-0/040O0s0w0$0T0,0.0:0=0@0_0{0}0 11131517190F0y0L0s0f0s1N0n1c1e1g1i0D1a1O1Q1S0y1U1m1o1q1A040%0*1z0/0s0;0?0^0`0|0~10121416181!0L0k1M1(1V1f0o0n1h1j221$1T0k1*1p1r0K1t0s0b000l0N0D0v0}0l0+2j1?1C1_1F1|1I1 1L2d1R2f1d0s2523251U1n2i1-0s0l0A0d0G0M0D0Y0s0C2v1A2x1^1E1{1H1~1K0j0h2I0h292O1,1s0B0U0r0}2#1=1@1D1`1G1}1J202/1e0J2?2j1.0r0d2}1B2(312B2,352I0L391-1u0h0r0E0E0e3e2%302A2+341L3n1s0%0)0s0K0#3G0d0v0v0e2_1;3f3x2*332D2.3l3C2k2_0g0#0m3v2 2z3S2C2-190J273m1+3a1u0a1p0Q.