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Math Help - Separable Ordinary D.E (help verify my working)

  1. #1
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    Separable Ordinary D.E (help verify my working)

    Hi everyone

    Need help to verify my working, thank you in advance for all help & support.

    Solve the following separable ordinary D.E.

    a) \frac{dy}{dx}=(5y-2)sinx
    \frac{dy}{5y-2}=sinxdx
    \int\frac{dy}{5y-2}=\int sinxdx+c
    \frac{1}{5}ln(5y-2)=-cosx+c

    b) \frac{dy}{dx}=\frac{x^2}{y+4}
    dy(y+4)=dx(x^2)
    \int (y+4)dy=\int (x^2)dx+c
    \frac{y^2}{2}+4y=\frac{x^3}{3}+c

    c) \frac{dy}{dx}=e^{x+y}
    \frac{dy}{dx}=e^xe^y
    e^{-y}dy=e^{x}dx
    \int e^{-y}dy=\int e^{x}dx+c
    -e^{-y}=e^x+c

    d) xy\frac{dy}{dx}=x^2+y^2,Hint:substitute u=\frac{y}{x}
    \frac{dy}{dx}=\frac{x}{y}+\frac{y}{x}
    u=\frac{y}{x},y=ux
    \frac{dy}{dx}=u+x\frac{du}{dx}
    u+x\frac{du}{dx}=\frac{1}{u}+u
    udu=\frac{1}{x}dx
    \int udu=\int \frac{1}{x}dx+c
    \frac{u^2}{2}=ln|x|+c
    \frac{y^2}{2x^2}=ln|x|+c

    Thank you in advance for all your kind help & support.
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  2. #2
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    Quote Originally Posted by anderson View Post
    Hi everyone

    Need help to verify my working, thank you in advance for all help & support.

    Solve the following separable ordinary D.E.

    a) \frac{dy}{dx}=(5y-2)sinx
    \frac{dy}{5y-2}=sinxdx
    \int\frac{dy}{5y-2}=\int sinxdx+c
    \frac{1}{5}{\color{red}ln(5y-2)}=-cosx+c

    b) \frac{dy}{dx}=\frac{x^2}{y+4}
    dy(y+4)=dx(x^2)
    \int (y+4)dy=\int (x^2)dx+c
    \frac{y^2}{2}+4y=\frac{x^3}{3}+c

    c) \frac{dy}{dx}=e^{x+y}
    \frac{dy}{dx}=e^xe^y
    e^{-y}dy=e^{x}dx
    \int e^{-y}dy=\int e^{x}dx+c
    -e^{-y}=e^x+c

    d) xy\frac{dy}{dx}=x^2+y^2,Hint:substitute u=\frac{y}{x}
    \frac{dy}{dx}=\frac{x}{y}+\frac{y}{x}
    u=\frac{y}{x},y=ux
    \frac{dy}{dx}=u+x\frac{du}{dx}
    u+x\frac{du}{dx}=\frac{1}{u}+u
    udu=\frac{1}{x}dx
    \int udu=\int \frac{1}{x}dx+c
    \frac{u^2}{2}=ln|x|+c
    \frac{y^2}{2x^2}=ln|x|+c

    Thank you in advance for all your kind help & support.
    Dear anderson,

    In the first problem (which I have highlighted) you have'nt written the modulus sign for the natural logarithm. Other than this all the other solutions are perfect.
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by anderson View Post
    Hi everyone

    Need help to verify my working, thank you in advance for all help & support.

    Solve the following separable ordinary D.E.

    a) \frac{dy}{dx}=(5y-2)sinx
    \frac{dy}{5y-2}=sinxdx
    \int\frac{dy}{5y-2}=\int sinxdx+c.................................................. (a)
    \frac{1}{5}ln(5y-2)=-cosx+c

    b) \frac{dy}{dx}=\frac{x^2}{y+4}
    dy(y+4)=dx(x^2)
    \int (y+4)dy=\int (x^2)dx+c...................................(b)
    \frac{y^2}{2}+4y=\frac{x^3}{3}+c

    c) \frac{dy}{dx}=e^{x+y}
    \frac{dy}{dx}=e^xe^y
    e^{-y}dy=e^{x}dx
    \int e^{-y}dy=\int e^{x}dx+c.................................................. ........(c)
    -e^{-y}=e^x+c

    d) xy\frac{dy}{dx}=x^2+y^2,Hint:substitute u=\frac{y}{x}
    \frac{dy}{dx}=\frac{x}{y}+\frac{y}{x}
    u=\frac{y}{x},y=ux
    \frac{dy}{dx}=u+x\frac{du}{dx}
    u+x\frac{du}{dx}=\frac{1}{u}+u
    udu=\frac{1}{x}dx
    \int udu=\int \frac{1}{x}dx+c.............................................(d)
    \frac{u^2}{2}=ln|x|+c
    \frac{y^2}{2x^2}=ln|x|+c

    Thank you in advance for all your kind help & support.
    Remove the +c from the lines (a),(b),(c),(d). The constant is added after you integrate, not before integration.
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