Aidez moi Please?

Question

a)Solve the differential equation using K as your arbitrary constant
y' = xy
b) 3. [SCalcCC2 7.3.08.]
Solve the differential equation. Let C represent an arbitrary constant.
dz/dt+exp(t+z)=0
c)Find the solution of the differential equation that satisfies the given initial condition
(x+2ysqrt(x^2+1))(dy/dx)=0, y(0)=1

Answer 1

(a) y' = dy/dx

y' - xy = 0

p(x) = -x
u(x) = exp(integral(p(x)dx)) = exp(-x^2/2 + c) = C * exp(-0.5 * x^2)

y(x) = K * exp(0.5 * x^2)

(b)

Answer 2

a) Split variables, then integrate.

[1] .. y' / y = x
[2] .. 1/y dy = x dx
[3] .. ln |y| = x^2/2 + K
[4] .. |y| = e^{x^2/2 + K}
[5] .. y = K' * e^{x^2/2}

b) Same method.

[6] .. dz/dt = -e^t * e^z
[7] .. dz/e^z = -dt e^t
[8] .. dz e^(-z) = -dt e^t
[9] .. -e^(-z) = -e^t + C
[10] .. z = -ln (e^t - C)

c) You sure the formula is correct? If y(0) = 1, the factor on the left is non-zero, therefore dy/dx = 0, so y is constant.

Answer 3

omg...i dont even know...
good luck man