eingehen
to die
Describes dying or withering, especially of plants or animals, often because of poor conditions such as lack of water or disease.
Ohne Wasser wird die Pflanze bald eingehen.
Without water, the plant will soon die.

to die / shrink / respond
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to die
Describes dying or withering, especially of plants or animals, often because of poor conditions such as lack of water or disease.
Ohne Wasser wird die Pflanze bald eingehen.
Without water, the plant will soon die.

'Eingehen' is a separable verb. The prefix 'ein-' goes to the end in a main clause. In subordinate clauses and in the infinitive, the verb stays together.
Die Blumen gehen ohne Sonne ein.
The flowers die without sun.
The verb 'eingehen' always forms the perfect tense with the auxiliary verb 'sein', regardless of the meaning. This is because of the basic meaning of 'gehen' (to go), which expresses a change of state or movement.
Meine Lieblingspflanze ist leider eingegangen.
My favorite plant has unfortunately died.
In the meaning of 'to respond' or 'to deal with', 'eingehen' is always used with the preposition 'auf' and the accusative. The structure is: auf jemanden/etwas eingehen (to respond to someone/something).
Der Redner ist auf die Fragen aus dem Publikum eingegangen.
The speaker responded to the questions from the audience.
The meaning 'to arrive' or 'to come in' is used very frequently in formal and business contexts. Typical subjects are 'eine Bestellung' (an order), 'eine Zahlung' (a payment), 'eine E-Mail' (an email), or 'eine Bewerbung' (an application).
Sobald die Bewerbung eingeht, erhalten Sie eine Bestätigung.
As soon as the application comes in, you will receive a confirmation.
When someone begins an obligation, 'eingehen' is often used with an accusative object. Common combinations are 'eine Beziehung eingehen' (to enter into a relationship), 'einen Vertrag eingehen' (to enter into a contract), 'ein Risiko eingehen' (to take a risk), or 'einen Kompromiss eingehen' (to make a compromise).
Beide Parteien müssen bereit sein, einen Kompromiss einzugehen.
Both parties must be willing to make a compromise.
The meanings 'to die' and 'to shrink' become clear through context. 'Eingehen' refers to plants, or sometimes animals, when it is about dying. It refers to clothing or fabrics when it is about shrinking.
Die Tomatenpflanzen sind in der Kälte eingegangen.
The tomato plants died in the cold.
The expression 'in die Geschichte eingehen' (to go down in history) is a fixed idiom. It means that an event or person becomes historically significant and remains remembered.
Die Erfindung des Internets wird in die Geschichte eingehen.
The invention of the internet will go down in history.
to begin
'beginnen' and 'eingehen' can both express that something new is starting. But 'beginnen' is general: you can begin a piece of work, a conversation, a course, or a text. In this relation, 'eingehen' only fits in fixed expressions such as 'eine Beziehung eingehen' (to enter into a relationship), 'einen Vertrag eingehen' (to enter into a contract), or 'ein Risiko eingehen' (to take a risk). This is not only about the beginning, but also about a bond, obligation, or risk. Learners may confuse the words because in both cases something starts. As a practical rule: use 'beginnen' for normal beginnings; with contracts, relationships, and risks, 'eingehen' sounds more natural.

Vor dem Umzug möchte sie keinen neuen Vertrag eingehen.
Before moving, she does not want to enter into a new contract.
Nach der kurzen Pause beginnt der Chef mit der Planung für den Sommer.
After the short break, the boss begins planning for the summer.