**Question**

**Introduction**

This
question was taken from a Facebook group.
The suggested substitution was added in to make it accessible to students
taking H2 Mathematics. In the original
question, no suggested substitution was given.

**Strategy**

Observe
that the given equation has

*x*and*y*. After substitution, we should get an equation with*v*and*x*only. How to make*y*“disappear”? One way is recognise patches that can be substituted for*v*. Another way is to make*y*the subject and differentiate that with respect to*x*. Replace the derivative^{dy}/_{dx}with your new expression. Here, I do*both*at once!**Solution**

**Remarks**

Usually we try to express

*y*in terms of*x*, but here, it is more convenient to express*x*in terms of*y*.
Since

*A*is an arbitrary constant,*A*+1 is still an arbitrary constant. Also, whenever there is a “ln” appearing in the solution, it is good to introduce “ln” with an arbitrary constant. So we can lump*A*and 1 together and call it ln*B*. Nothing is lost in this process. ln*B*is able to achieve all possible numbers. ln*B*is negative if*B*is a fraction between 0 and 1, and is positive if*B*is more than 1 and is zero if*B*= 1.
This is not
an applied differential equation, so there are no exogenic reasons or hints for
us to suppose that

*x*+*y*+ 2 is a positive quantity. As such, we still need to leave it as |*x*+*y*+ 2|. To get rid of the absolute or modulus sign properly, we can replace ±*B*with*C*.
H04. Look for
pattern(s)

H05. Work
backwards

H09. Restate
the problem in another way

H10. Simplify
the problem

H11. Solve part
of the problem

H13. Use
Equation / write a Mathematical Sentence

__Heuristic that cannot be used here__
H08. Make
suppositions

**Suitable Levels**

*****GCE ‘A’ Level H2 Mathematics

*****IB HL Mathematics Calculus Option

*****Advanced Placement (AP)

* university
/ college calculus

* other
syllabuses that involve differential equations

* anyone
who is game for a challenge

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