Question #103557

Solve the following ordinary differential equations

(a) dy/dx+ycotx=power of e is cosx

For x=π/2 y=-2

(b) d square y/dx square + dy/dx + y =0

(a) dy/dx+ycotx=power of e is cosx

For x=π/2 y=-2

(b) d square y/dx square + dy/dx + y =0

Expert's answer

a)

"p=\\cot x, q=e^{cosx}"

"IF=\\exp\\int pdx=\\exp\\int \\cot xdx=\\exp (\\ln\\sin x)"

"IF=\\sin x"

"y\\sin x=\\int e^{cosx}\\sin x dx=c-e^{cosx}"

"-2\\sin 90=c-e^{cos90}\\to c=-1"

"y=-\\frac{1+e^{cos90}}{\\sin x}"

b)

"\\lambda ^2+\\lambda +1=0"

"\\lambda_{1,2}=\\frac{-1\\pm \\sqrt{3}}{2}"

"y=c_1e^{-\\frac{x}{2}}\\sin\\left (\\frac{\\sqrt{3}}{2}x\\right)+c_2e^{-\\frac{x}{2}}\\cos\\left (\\frac{\\sqrt{3}}{2}x\\right)"

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