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Solve $aU_x+bU_y+cU$=0 using characteristic method. I know how to solve this by change of coordinates as in this article.

But without changing coordinates how to do it with the method of characteristics? I know how to solve if it was $aU_x+bU_y=0 $by using directional derivative.

Also in page 10 of that article they have used Jacobian.Please explain why Jacobian is used there.

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Example 3 on page 16 is exactly this PDE solved using the method of characteristics. Please specify what is unclear about this solution.

As for the Jacobian, it is used on page 10 while they are still doing coordinate transformation. They look at the Jacobian to justify that the coordinate change is invertible. The Jacobian is not actually involved in obtaining a formula for the solution. And it does not appear when the method of characteristics is used.

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