dsolve gives wrong equations?

hey all!

so i'm trying to learn how dsolve and I know i'm doing something incorrectly but i don't know exactly what.

it could just be that everything is in order but i am just bad at math lol.

anyway, examples are below/ would love some help.

>> syms y(x)
>> dsolve(diff(y)==y+sin(x))
ans =
 C1*exp(x) - (2^(1/2)*cos(x - pi/4))/2 %should'nt the correct answer be -1/2*cos(t)-1/2*sin(t)+exp(t)*C1?
 
 
 e.g 2:
 syms y(x)
>> dsolve((diff(y))^2+y^2==1)
ans =
(exp(C1*1i - x*1i)*(exp(- C1*2i + x*2i) + 1))/2  %shoud'nt these two be sin(s-C1) and -sin(s-C1)?
(exp(C2*1i + x*1i)*(exp(- C2*2i - x*2i) + 1))/2  
                                              1  %legit
                                             -1  %legit
                                             

Answer 1

[Torsten]

There are often several ways to write a function. But it seems to me that your S2 solutions are not independent.

syms y(x) C1
S1 = dsolve(diff(y)==y+sin(x))

S1 = 

S2 = -1/2*cos(x)-1/2*sin(x)+exp(x)*C1

S2 = 

res1 = simplify(diff(S1,x)-S1-sin(x))

res1 = 

0

res2 = simplify(diff(S2,x)-S2-sin(x))

res2 = 

0

S1 = dsolve((diff(y))^2+y^2==1)

S1 = 

S2 = [sin(x-C1) ;-sin(x-C1)]

S2 = 

res1 = simplify(diff(S1,x).^2+S1.^2-1)

res1 = 

res2 = simplify(diff(S2,x).^2+S2.^2-1)

res2 = 

[daniel slama]

damn, TYSM.

i never would have figured this out on my own