Euler's and picard's

 &#3627408386;&#3627408424;&#3627408423;&#3627408428;&#3627408418;&#3627408413;&#3627408414;&#3627408427; &#3627408429;&#3627408417;&#3627408414; &#3627408416;&#3627408414;&#3627408423;&#3627408414;&#3627408427;&#3627408410;&#3627408421; &#3627408415;&#3627408418;&#3627408427;&#3627408428;&#3627408429; &#3627408424;&#3627408427;&#3627408413;&#3627408414;&#3627408427; &#3627408413;&#3627408418;&#3627408415;&#3627408415;&#3627408414;&#3627408427;&#3627408414;&#3627408423;&#3627408429;&#3627408418;&#3627408410;&#3627408421; ??????&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423;
&#3627408517;&#3627408538;
&#3627408517;&#3627408537;
= &#3627408519; &#3627408537;,&#3627408538; .−−−−−− &#3627409359; &#3627408432;&#3627408418;&#3627408429;&#3627408417; &#3627408429;&#3627408417;&#3627408414; &#3627408418;&#3627408423;&#3627408418;&#3627408429;&#3627408418;&#3627408410;&#3627408421; &#3627408412;&#3627408424;&#3627408423;&#3627408413;&#3627408418;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408538; &#3627408537;
&#3627409358; = &#3627408538;
&#3627409358;
 &#3627408403;&#3627408417;&#3627408414; &#3627408414;&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627409359; &#3627408412;&#3627408410;&#3627408423; &#3627408411;&#3627408414; &#3627408432;&#3627408427;&#3627408418;&#3627408429;&#3627408414; &#3627408410;&#3627408428;
&#3627408517;&#3627408538;=&#3627408519; &#3627408537;,&#3627408538; &#3627408517;&#3627408537;
 &#3627408418;&#3627408423;&#3627408429;&#3627408414;&#3627408416;&#3627408427;&#3627408410;&#3627408429;&#3627408418;&#3627408423;&#3627408416; &#3627408424;&#3627408423; &#3627408411;&#3627408424;&#3627408429;&#3627408417; &#3627408428;&#3627408418;&#3627408413;&#3627408414;&#3627408428; &#3627408432;&#3627408418;&#3627408429;&#3627408417; &#3627408421;&#3627408418;&#3627408422;&#3627408418;&#3627408429;&#3627408428; &#3627408415;&#3627408424;&#3627408427; &#3627408537; &#3627408410;&#3627408423;&#3627408413; &#3627408538;,&#3627408432;&#3627408414; &#3627408416;&#3627408414;&#3627408429;
&#3627408517;&#3627408538;
&#3627408538;
&#3627408538;
&#3627409358;
= &#3627408519; &#3627408537;,&#3627408538; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;
&#3627409358;

(&#3627408538;−&#3627408538;
&#3627409358;)= &#3627408519; &#3627408537;,&#3627408538; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;&#3627409358;

&#3627408538;=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;&#3627409358;
−−−−−(&#3627409360;)
??????&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; (2) is know as integral equation and can be solved by successive Approximation.
MANIKANTA SATYALA

 &#3627408397;&#3627408424;&#3627408432; &#3627408429;&#3627408417;&#3627408414; &#3627409359;
&#3627408428;&#3627408429;
&#3627408384;&#3627408425;&#3627408425;&#3627408427;&#3627408424;&#3627408433;&#3627408418;&#3627408422;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408418;&#3627408423; ??????&#3627408418;&#3627408412;&#3627408424;&#3627408427;&#3627408413;
′
&#3627408428; &#3627408396;&#3627408414;&#3627408429;&#3627408417;&#3627408424;&#3627408413;
&#3627408385;&#3627408434; &#3627408427;&#3627408414;&#3627408425;&#3627408421;&#3627408410;&#3627408412;&#3627408418;&#3627408423;&#3627408416; &#3627408538; &#3627408411;&#3627408434; &#3627408434;
&#3627409358; &#3627408418;&#3627408423; &#3627408519; &#3627408537;,&#3627408538; &#3627408418;&#3627408423; &#3627408401;.&#3627408391;.&#3627408402; &#3627408424;&#3627408415; &#3627408414;&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627409360; ,
&#3627408432;&#3627408414; &#3627408416;&#3627408414;&#3627408429;
&#3627408538;
&#3627409359;=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538;
&#3627409358; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;&#3627409358;

 &#3627408397;&#3627408424;&#3627408432; &#3627408429;&#3627408417;&#3627408414; &#3627409360;
&#3627408423;&#3627408413;
&#3627408384;&#3627408425;&#3627408425;&#3627408427;&#3627408424;&#3627408433;&#3627408418;&#3627408422;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408418;&#3627408423; ??????&#3627408418;&#3627408412;&#3627408424;&#3627408427;&#3627408413;
′
&#3627408428; &#3627408396;&#3627408414;&#3627408429;&#3627408417;&#3627408424;&#3627408413;
&#3627408385;&#3627408434; &#3627408427;&#3627408414;&#3627408425;&#3627408421;&#3627408410;&#3627408412;&#3627408418;&#3627408423;&#3627408416; &#3627408538; &#3627408411;&#3627408434; &#3627408434;
&#3627409359; &#3627408418;&#3627408423; &#3627408519; &#3627408537;,&#3627408538; &#3627408418;&#3627408423; &#3627408401;.&#3627408391;.&#3627408402; &#3627408424;&#3627408415; &#3627408414;&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627409360; ,
&#3627408432;&#3627408414; &#3627408416;&#3627408414;&#3627408429;
&#3627408538;
&#3627409360;=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538;
&#3627409359; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;&#3627409358;

 &#3627408397;&#3627408424;&#3627408432; &#3627408429;&#3627408417;&#3627408414; &#3627408423;
&#3627408429;&#3627408417;
&#3627408384;&#3627408425;&#3627408425;&#3627408427;&#3627408424;&#3627408433;&#3627408418;&#3627408422;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408418;&#3627408423; ??????&#3627408418;&#3627408412;&#3627408424;&#3627408427;&#3627408413;
′
&#3627408428; &#3627408396;&#3627408414;&#3627408429;&#3627408417;&#3627408424;&#3627408413;
&#3627408385;&#3627408434; &#3627408427;&#3627408414;&#3627408425;&#3627408421;&#3627408410;&#3627408412;&#3627408418;&#3627408423;&#3627408416; &#3627408538; &#3627408411;&#3627408434; &#3627408434;
&#3627408423;−&#3627409359; &#3627408418;&#3627408423; &#3627408519; &#3627408537;,&#3627408538; &#3627408418;&#3627408423; &#3627408401;.&#3627408391;.&#3627408402; &#3627408424;&#3627408415; &#3627408414;&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627409360; ,
&#3627408432;&#3627408414; &#3627408416;&#3627408414;&#3627408429;
&#3627408538;
??????=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538;
??????−&#3627409359; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;
&#3627409358;

&#3627408402;&#3627408429;&#3627408424;&#3627408425; &#3627408429;&#3627408417;&#3627408414; &#3627408425;&#3627408427;&#3627408424;&#3627408412;&#3627408414;&#3627408428;&#3627408428; &#3627408432;&#3627408417;&#3627408414;&#3627408423; &#3627408434;&#3627408424;&#3627408430; &#3627408416;&#3627408414;&#3627408429; &#3627408429;&#3627408432;&#3627408424; &#3627408412;&#3627408424;&#3627408423;&#3627408428;&#3627408414;&#3627408412;&#3627408430;&#3627408429;&#3627408418;&#3627408431;&#3627408414; &#3627408431;&#3627408410;&#3627408421;&#3627408430;&#3627408414;&#3627408428; &#3627408424;&#3627408415; &#3627408538;

MANIKANTA SATYALA

MANIKANTA SATYALAExample: Using Picard
′
s method solve
??????&#3627408486;
??????&#3627408485;
=3+2&#3627408485;&#3627408486; where &#3627408486; 0 =1 for &#3627408485;=0.1
Solution: By Picard
′
s Method &#3627408538;
??????=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538;
??????−&#3627409359; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;
&#3627409358;

Here &#3627408485;
0=0,&#3627408486;
0=1,?????? &#3627408485;,&#3627408486; =3+2&#3627408485;&#3627408486;
1
st
Approximation:
Put ??????=1 and &#3627408486;
0=1 in ?????? &#3627408485;,&#3627408486;
&#3627408538;
&#3627409359;=&#3627408538;
&#3627409358;+ &#3627408519; &#3627408537;,&#3627408538;
&#3627409358; &#3627408517;&#3627408537;
&#3627408537;
&#3627408537;&#3627409358;

∴ &#3627408486;
1=1+3&#3627408485;+&#3627408485;
2
=1.31(???????????? &#3627408485;=0.1)
2
nd
Approximation:
Put ??????=2 and &#3627408486;
1=1+3&#3627408485;+&#3627408485;
2
in ?????? &#3627408485;,&#3627408486;
&#3627408538;
&#3627409360;=&#3627409359;+ &#3627409361;+&#3627409360;&#3627408537; &#3627409359;+&#3627409361;&#3627408537;+&#3627408537;
&#3627409360;
&#3627408517;&#3627408537;
&#3627408537;
&#3627409358;

∴ &#3627408486;
2=1+3&#3627408485;+&#3627408485;
2
+2&#3627408485;
3
+
&#3627408485;
4
2
=1.31205(???????????? &#3627408485;=0.1)
This is the approximation solution

MANIKANTA SATYALA &#3627408386;&#3627408424;&#3627408423;&#3627408428;&#3627408418;&#3627408413;&#3627408414;&#3627408427; &#3627408429;&#3627408417;&#3627408414; &#3627408416;&#3627408414;&#3627408423;&#3627408414;&#3627408427;&#3627408410;&#3627408421; &#3627408415;&#3627408418;&#3627408427;&#3627408428;&#3627408429; &#3627408424;&#3627408427;&#3627408413;&#3627408414;&#3627408427; &#3627408413;&#3627408418;&#3627408415;&#3627408415;&#3627408414;&#3627408427;&#3627408414;&#3627408423;&#3627408429;&#3627408418;&#3627408410;&#3627408421; ??????&#3627408426;&#3627408430;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423;
&#3627408517;&#3627408538;
&#3627408517;&#3627408537;
= &#3627408519; &#3627408537;,&#3627408538; , &#3627408432;&#3627408418;&#3627408429;&#3627408417; &#3627408429;&#3627408417;&#3627408414; &#3627408418;&#3627408423;&#3627408418;&#3627408429;&#3627408418;&#3627408410;&#3627408421; &#3627408412;&#3627408424;&#3627408423;&#3627408413;&#3627408418;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408538; &#3627408537;
&#3627409358; = &#3627408538;
&#3627409358;
 &#3627408392;&#3627408415; &#3627408538; &#3627408537; &#3627408418;&#3627408428; &#3627408429;&#3627408417;&#3627408414; &#3627408414;&#3627408433;&#3627408410;&#3627408412;&#3627408429; &#3627408428;&#3627408424;&#3627408421;&#3627408430;&#3627408429;&#3627408418;&#3627408424;&#3627408423; &#3627408424;&#3627408415;
&#3627408517;&#3627408538;
&#3627408517;&#3627408537;
&#3627408429;&#3627408417;&#3627408414;&#3627408423; &#3627408429;&#3627408417;&#3627408414; &#3627408403;&#3627408410;&#3627408434;&#3627408421;&#3627408424;&#3627408427;’&#3627408428; &#3627408428;&#3627408414;&#3627408427;&#3627408418;&#3627408414;&#3627408428; &#3627408415;&#3627408424;&#3627408427; &#3627408538;(&#3627408537;) &#3627408418;&#3627408428;
&#3627408538; &#3627408537; = &#3627408538;(&#3627408537;
&#3627409358;)+
&#3627408537; − &#3627408537;
&#3627409358;
&#3627409359;!
&#3627408538;′ &#3627408537;
&#3627409358;
+
&#3627408537; − &#3627408537;
&#3627409358;
&#3627409360;
&#3627409360;!
&#3627408538;′′ &#3627408537;
&#3627409358;
+ ...− − − &#3627409359;
 &#3627408397;&#3627408424;&#3627408432; &#3627408428;&#3627408430;&#3627408411;&#3627408428;&#3627408429;&#3627408418;&#3627408429;&#3627408430;&#3627408429;&#3627408418;&#3627408423;&#3627408416; &#3627408521; = &#3627408537;
&#3627409359; – &#3627408537;
&#3627409358; &#3627408418;&#3627408423; &#3627408414;&#3627408426; &#3627409359; ,&#3627408432;&#3627408414; &#3627408416;&#3627408414;&#3627408429;
&#3627408538; &#3627408537;
&#3627409359; = &#3627408538; &#3627408537;
&#3627409358; + &#3627408521;&#3627408538;′ &#3627408537;
&#3627409358; +
&#3627408521;
&#3627409360;
&#3627409360;!
&#3627408538;′′(&#3627408537;
&#3627409358;) + ...
 &#3627408392;&#3627408415; &#3627408521; &#3627408418;&#3627408428; &#3627408412;&#3627408417;&#3627408424;&#3627408428;&#3627408414;&#3627408423; &#3627408428;&#3627408422;&#3627408410;&#3627408421;&#3627408421; &#3627408414;&#3627408423;&#3627408424;&#3627408430;&#3627408416;&#3627408417; &#3627408429;&#3627408417;&#3627408414;&#3627408423; &#3627408432;&#3627408414; &#3627408422;&#3627408410;&#3627408434; &#3627408423;&#3627408414;&#3627408416;&#3627408421;&#3627408414;&#3627408412;&#3627408429; &#3627408429;&#3627408417;&#3627408414; &#3627408428;&#3627408414;&#3627408412;&#3627408424;&#3627408423;&#3627408413; &#3627408410;&#3627408423;&#3627408413; &#3627408417;&#3627408418;&#3627408416;&#3627408417;&#3627408414;&#3627408427; &#3627408424;&#3627408427;&#3627408413;&#3627408414;&#3627408427; &#3627408429;&#3627408414;&#3627408427;&#3627408422; &#3627408424;&#3627408415; &#3627408521;.
&#3627408538;
&#3627409359; = &#3627408538;
&#3627409358;+&#3627408521;&#3627408519; &#3627408537;
&#3627409358;,&#3627408538;
&#3627409358;

??????&#3627408417;&#3627408418;&#3627408412;&#3627408417; &#3627408418;&#3627408428; ??????&#3627408430;&#3627408421;&#3627408414;&#3627408427;’&#3627408428; &#3627408415;&#3627408418;&#3627408427;&#3627408428;&#3627408429; &#3627408410;&#3627408425;&#3627408425;&#3627408427;&#3627408424;&#3627408433;&#3627408418;&#3627408422;&#3627408410;&#3627408429;&#3627408418;&#3627408424;&#3627408423;.
 &#3627408403;&#3627408417;&#3627408414; &#3627408416;&#3627408414;&#3627408423;&#3627408414;&#3627408427;&#3627408410;&#3627408421; &#3627408428;&#3627408429;&#3627408414;&#3627408425; &#3627408415;&#3627408424;&#3627408427; ??????&#3627408430;&#3627408421;&#3627408414;&#3627408427; &#3627408422;&#3627408414;&#3627408429;&#3627408417;&#3627408424;&#3627408413; &#3627408418;&#3627408428;
&#3627408538;
&#3627408522;+&#3627409359;=&#3627408538;
&#3627408522; + &#3627408521;&#3627408519;(&#3627408537;
&#3627408522; ,&#3627408538;
&#3627408522;) &#3627408536;&#3627408521;&#3627408518;??????&#3627408518; &#3627408522; = &#3627409358;,&#3627409359;,&#3627409360;....

MANIKANTA SATYALAProblem: Using Euler’s method,find &#3627408486; 0.2 ,given by
??????&#3627408486;
??????&#3627408485;
=&#3627408486;−
2&#3627408485;
&#3627408486;
,&#3627408486; 0 =1 and ℎ=0.1
Solution: Here &#3627408485;
0=0,&#3627408486;
0=1 and
??????&#3627408486;
??????&#3627408485;
=?????? &#3627408485;,&#3627408486; =&#3627408486;−
2&#3627408485;
&#3627408486;
also ℎ=0.1
1
st
Approximation: -
&#3627408486;
1=&#3627408486;
0+ℎ ?????? &#3627408485;
0,&#3627408486;
0
=1+ 0.1 1−0 =1.1
&#3627408485;
1=&#3627408485;
0+ℎ=0+0.1=0.1
2
nd
Approximation: -
&#3627408486;
2=&#3627408486;
1+ℎ ?????? &#3627408485;
1,&#3627408486;
1
=1.1+ 0.1 0.1−
2 0.1
1.1
=1.1918
&#3627408485;
2=&#3627408485;
1+ℎ=01+0.1=0.2
∴&#3627408486; 0.2 =1.1918

MANIKANTA SATYALA

Euler's and picard's

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Euler's and picard's