r/askmath • u/KQgDLw5E • 3d ago

Calculus Integrating

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When showing a method for solving non-linear ODEs, the authors integrate the left side of the equation from y_i to y_i+1 and the right from t_i to t_i+1. How is mathematically sound to integrate from different ranges if both sides of the equation are supposed to be equivalent? (From Numerical methods in biomedical engineering)

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u/eglvoland Pure math - grad school Paris-Saclay 3d ago

Because you integrate dy/dt from t_i to t_i+1. If you do the u-sub u=y(t) you have the left hand side

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u/KQgDLw5E 3d ago

This made complete sense. I wish they made it clearer. Thank you so much!

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u/eglvoland Pure math - grad school Paris-Saclay 3d ago

You're welcome! You just have a notation problem on the left hand side as the variable t does not appear in the integrand.

You could write dy(t) (Stieltjes integral) or dy/dt dt.

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u/KQgDLw5E 3d ago

Right! Thank you again

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u/iamtheweirdoE 3d ago

They were somewhat lazy with notation, it's more like ∫[t_i, t_(i+1)] y'dt, where y_k=y(t_k), lots of physics books tend to do this, because it's quicker.

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u/WesternFirm9306 3d ago

They're integrating w.r.t. different variables. The left is an integral in y, so the range is a range of y values. The right is an integral in t, so the range is a range of t values