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The function is .
Error function is .
Consider .
Differentiate on each side with respect to .
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Product rule of derivatives: .
Substitute in above expression.
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Find the derivative of the error function.
Consider .
Fundamental theorem of calculus :
If is continuous on , then the function is defined by is continuous on and differentiable on , then .
Substitute in above expression.
Therefore the function satisfies the differential equation .
The function satisfies the differential equation .
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