dy/dx=x^2+y^2; y(0)=0(adsbygoogle = window.adsbygoogle || []).push({});

okay i solved this question using homogenous

dy/dx=1+(y/x)^2

subsitute v=y/x

v+x*dv/dx=1+v^2 then use exact equation

dv/(v^2-v+1)=dx/x

dv/((v-1/2)^2+3/4)=(dx/x or just ln(x))

then integrate dv/((v-1/2)^2+3/4)

u=v-1/2

du=dv

dv/(u^2)+3/4

s=2u/sqrt(3)

ds=2/sqrt(3)du

thus

2/sqrt[3]*arctan=ln[x]

then substitue everything back in

2/sqrt[3]*arctan[2u/sqrt[3]]=ln[x]+c

2/sqrt[3]*arctan[2(v-1/2)/sqrt[3]]=ln[x]+c

2/sqrt[3]*arctan[2(y/x-1/2)/sqrt[3]]=ln[x]+c

okay ths thing is now i can not plug in y(0)=0 to get the constant for c and the other thing is this is suppose to be a unique and exisit throughout the interval [0,1/2] i used picard's theorem to show that its continus and estimated h=[0,1/2] the thing is im suppose to plug in 1/2 into y

y(1/2) and when i solve the equation however i can't get a constant

and when i use picard's iteration method successful approximation i do not get the same number. well one thing is i don't have the constant c unless i did the ODE totally wrong. can't think of another first order technique.

when i do picard's iteration method i get .047**** but if i have c i could might have the same answer when i plug in 1/2 one question is am i suppose to do solving it by subtitution and how is it possible tof ind the constant term? or am i totally wrong?

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# 1st order ODE Question

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