$\"\"$

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The function is $\"\"$ .

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Error function is $\"\"$ .

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Consider $\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$ .

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Product rule of derivatives: $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$ .

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$\"\"$

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Find the derivative of the error function.

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Consider $\"\"$ .

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Fundamental theorem of calculus :

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If $\"\"$ is continuous on $\"\"$ , then the function $\"\"$ is defined by $\"\"$ is continuous on $\"\"$ and differentiable on $\"\"$ , then $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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$\"\"$

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Therefore the function satisfies the differential equation $\"\"$ .

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$\"\"$

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The function satisfies the differential equation $\"\"$ .