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Verify Solutions Of D Es Exercises Worksheet

Verify that

{$$ \large y=\left(1+ce^{-x^2/2}\;\right) \left(1-ce^{-x^2/2}\right)^{-1} $$}

is a solution of

{$$ \large 2y'+x(y^2-1)=0 $$}

on some interval, for any choice of the arbitrary constants appearing in the function

Trench, William F., "Elementary Differential Equations with Boundary Value Problems" (2013). Faculty Authored Books. 9. Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Some preliminaries to make things easier:

{$$ \begin{align} \left(1+ce^{-x^2/2}\;\right)^\prime &= 0+c({{-x^2}\over{2}})^\prime (e^{-x^2/2}\;)^\prime \cr &= c\left(-{1\over 2}\right)(2x)(e^{-x^2/2}\;) \cr &= -cxe^{-x^2/2} \cr \left(1-ce^{-x^2/2}\;\right)^\prime &= cxe^{-x^2/2}\end{align} $$}

{$$ \begin{align} y^2&= \left[\left(1+ce^{-x^2/2}\;\right) \left(1-ce^{-x^2/2}\;\right)^{-1}\right]^2 \cr &= \left(1+ce^{-x^2/2}\;\right)^2 \left(1-ce^{-x^2/2}\;\right)^{-2} \cr &= {{1+ 2ce^{-x^2/2} + ce^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} \end{align} $$}

{$$ \begin{align} y^\prime &= {(-cxe^{-x^2/2}\;) \left(1-ce^{-x^2/2}\;\right) -(cxe^{-x^2/2}\;)\left(1+ce^{-x^2/2}\;\right)\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} \cr &= {(-cxe^{-x^2/2}\;)- (-cxe^{-x^2/2}\;)(ce^{-x^2/2}\;) -\left((cxe^{-x^2/2}\;)+(cxe^{-x^2/2}\;)(ce^{-x^2/2}\;)\right)\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} \cr &= { {-cxe^{-x^2/2} + cxe^{-x^2} - cxe^{-x^2/2} - cxe^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} \cr &= { {-2cxe^{-x^2/2} }\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} \end{align} $$}

Now for the main course:

{$$ \begin{align} 2y'+x(y^2-1)&=0 \cr 2 { {-2cxe^{-x^2/2} }\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} + x {{1+ 2ce^{-x^2/2} + ce^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} -x \quad & ? \quad 0 \cr { {-4cxe^{-x^2/2} }\over{ \left(1-ce^{-x^2/2}\;\right)^2 }} + {{x+ 2cxe^{-x^2/2} + cxe^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} -x \quad & ? \quad 0 \cr { {x- 2cxe^{-x^2/2} + cxe^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} - {{x(1-ce^{-x^2/2})^{2}}\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} \quad & ? \quad 0 \cr { {x- 2cxe^{-x^2/2} + cxe^{-x^2} }\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} - {{(x-2cxe^{-x^2/2}-xce^{-x^2})}\over{ \left(1-ce^{-x^2/2}\;\right)^{2}}} \quad & ? \quad 0 \cr \therefore 0 &= 0\end{align} $$}


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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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