Solving 1st Order ODE with Homogeneous Techniques | Initial Value y(0)=0

[dglee]

dy/dx=x^2+y^2; y(0)=0

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 I am 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?

 

[mighty2000]

Hello! Thank you for sharing your solution to this problem. It looks like you have approached it correctly using the method of substitution and integrating to find the solution.

As for finding the constant term, you can use the initial condition given in the problem to solve for it. In this case, we know that y(0) = 0, so we can plug that into the equation to get:

2/sqrt[3]*arctan[2(0-1/2)/sqrt[3]]=ln[0]+c

-2/sqrt[3]*arctan[1/sqrt[3]]=c

c = -2/sqrt[3]*arctan[1/sqrt[3]]

So, your final solution would be:

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

As for the discrepancy in your Picard's iteration method, it could be due to rounding errors or a mistake in the iteration process. I would suggest double checking your calculations and making sure you are using the correct initial value for your first iteration.

Overall, it seems that you have a good understanding of how to approach this problem and have used the correct techniques. Keep up the good work!