[H2_20151125DEST] Differential Equation via Substitution

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.

Heuristics Used

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) Calculus BC

*  university / college calculus

*  other syllabuses that involve differential equations

*  anyone who is game for a challenge