Question

a)

Let y be the solution of the equation

y ′ = sqrt(1 − y^2) satisfying the condition  y ( 0 ) = 0.

Find the value of the function  f ( x ) = sqrt(2)*y ( x ) at x = π/4.

(The square root in the right hand side

of the equation takes positive values and − 1 ≤ y ≤ 1)

b)

Let y be the solution of the equation

y ′ = 5 x^4 sin ⁡(x^5) satisfying the condition y ( 0 ) = − 1.

Find y ( (π)^1/5 ).

c)

Find the largest value of the parameter r

for which the function y = e^(rx) is a solution of the

equation y ″ − 12 y ′ + 27 y = 0.

d)

Let y ′ = − 3x^2*e^(-x^3) and let y ( 0 ) = 1.

Find  ln ⁡ ( y ( 2 ) ).

e)

Find the smallest value of the parameter r

for which the function y = e^(rx) is a solution of the

equation y ″ − 12 y ′ + 27 y = 0.

Answer 1