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Verify Solutions of DEs Worked 3

Exercises

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Verify Solutions of DEs

Exercises

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.

7.

{$$\large \begin{multline}y=c_1e^x+c_2x+ {2\over x}\;; \cr \shoveleft (1-x)y''+xy'- y=4(1-x-x^2)x^{-3} \end{multline}$$}

{\begin{align} y&=c_1e^x+c_2x+ {2\over x} \cr y^\prime &= c_1e^x+c_2-{2\over {x^2}} \cr y^{\prime\prime} &= c_1e^x+{4\over {x^3}} \cr (1-x)y^{\prime\prime} &= c_1e^x +{4\over {x^3}} - c_1xe^x -{4\over {x^2}} \cr xy^\prime &= c_1xe^x +c_2x -{2\over {x}} \cr (1-x)y^{\prime\prime} + xy^\prime &= c_1e^x +{4\over {x^3}} - c_1xe^x -{4\over {x^2}} + c_1xe^x +c_2x -{2\over {x}} \cr \cr (1-x)y^{\prime\prime} + xy^\prime -y &= c_1e^x +{4\over {x^3}} - c_1xe^x -{4\over {x^2}} + c_1xe^x +c_2x -{2\over {x}} - c_1e^x-c_2x-{2\over x} \cr &= +{4\over {x^3}} -{4\over {x^2}} -{4\over x} \cr \therefore (1-x)y^{\prime\prime} + xy^\prime -y &= 4( 1 - x - x^2)x^{-3} \cr \end{align}}

8.

{$$\large \begin{multline}y=x^{-1/2}\;(c_1\sin x+c_2 \cos x)+4x+8\;; \cr \shoveleft x^2y''+xy'+ {\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2 \end{multline}$$}

First we should notice that

{$$y = 4x + 8$$}

is a solution, so it is only necessary to show the other terms come to nothing.

Let

{\large \begin{align} T_1 & = x^{-{{1}\over{2}}}c_1\sin x \end{align}}

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