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Math Help - Ordinary differential equations

  1. #1
    Junior Member ceode's Avatar
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    Ordinary differential equations

    1 ) SOLVE
    dy/dx + y*cot(x) =5*e^cos(x) if y=-4 and x=5/4

    2)
    solve

    dy/dx + (y logy)/x = y(logy)^2/(x^2)

    3)SOLVE
    (x*y^2)dx - e^(1/(x^3))dx - x^2*y dy = 0


    Thank you very much.
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  2. #2
    Junior Member ceode's Avatar
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    For clarity 3rd question is rewritten as

    SOLVE
    (x*y^2)dx - e^(1/(x^3))dx - (x^2)*y dy = 0
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  3. #3
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    Hello, ceode!

    \text{(1) Solve: }\;\dfrac{dy}{dx} + y\cdot\cot x  \:=\:5e^{\cos x},\quad y\left(\tfrac{5}{4}\right) \,=\,-4

    \text{Integrating factor: }\;I \:=\:e^{\int\cot x\,dx} \;=\;e^{\ln(\sin x)} \;=\;\sin x


    \text{Multiply by }I\!:\;\;\sin x\cdot\frac{dy}{dx} + y\cdot\cot x\cdot\sin x \;=\;5\sin x\cdot e^{\cos x}


    \text{We have: }\;\dfrac{d}{dx}\left(\sin x\cdot y\right) \;=\;5\sin x\cdot e^{\cos x}


    \displaystyle \text{Integrate: }\;\int\frac{d}{dx}(\sin x\cdot y) \;=\;5\int\sin x\cdot e^{\cos x}dx

    . . . . . . . . . . . . . . y\cdot \sin x \;=\;-5e^{\cos x} + C

    . . . . . . . . . . . . . . . . . . y \;=\;\dfrac{C - 5e^{\cos x}}{\sin x}


    y\left(\frac{5}{4}\right) \,=\,-4\!:\;\; \dfrac{C - 5e^{\cos\frac{5}{4}}}{\sin\frac{5}{4}}} \;=\;-4 \quad\Rightarrow\quad C \;=\;5e^{\cos\frac{5}{4}} - 4\sin\frac{5}{4}


    \text{Therefore: }\;y \;=\;\dfrac{5e^{\cos\frac{5}{4}} - 4\sin\frac{5}{4} - 5e^{\cos x}}{\sin x}
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  4. #4
    Junior Member ceode's Avatar
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    Thank you

    Thank you sir , can you help me with the other 2 questions
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