# HOW TO SOLVE DIFFERENTIAL EQUATION WORD PROBLEMS

## About the topic "How to solve differential equation word problems"

"How to solve differential equation word problems ?" is a big question having had by all the students who study quantitative aptitude to get prepared for competitive exams.For some students, solving differential equation word problems is never being easy and always it is a challenging one.

## How to solve differential equation word problems ?

The get answer  for the question "How to solve differential equation word problems ? "is purely depending upon the question that we have in the topic "Differential Equations". The techniques and methods we apply to solve word problems on differential equations will vary from problem to problem.

The techniques and methods we apply to solve a particular  word problem  will not work for another word problem in this topic.

Even though we have different techniques to solve word problems in different topics of math, let us see the stuff which is needed to solve any differential equation word problems.

## Stuff needed to solve differential equation word problems ## Steps involved in solving a differential equation word problem

To get answer for the question "How to solve differential equation word problems ? let us look at the steps involved in solving the  differential equation word problem given.

Problem:

In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant. At the end of one hour, 60 grams remain and at the end of 4 hours 21 grams. How many grams of the substance was there
initially?

Solution:

Step 1 :

Let us understand the given information. There are three information given in the question.

1. In a certain chemical reaction the rate of conversion of a substance at time "t" is proportional to the quantity of the substance still untransformed at that instant.

2. At the end of one hour, remaining substance is 60 grams.

3. At the end of 4 hours, remaining substance is 21 grams.

Step 2 :

Target of the question: How many grams of the substance

was there initially?

Step 3 :

Let "A" be the untransformed substance at any time "t"

Conversion of the substance "A" at time "t" can be written using differential coefficient as "dA/dt"

Step 4 :

Using the first information we have

dA/dt ∝ A  -----------> dA/dt = kA

dA/A = kdt

Step 5 :

To get "A", integrate the above equation on both the sides

(dA/A) =kdt ----------> log (A) = kt  + c

Taking "e" as base on both the sides, we have

elog(A) = ekt + c

A = ekt.ec

A = ekt.C

A = Cekt----------(1)

Step 6 :

Let us use the second and third information in (1).

That is

when t = 1, A = 60 --------> Cek = 60 ----- (2)

when t = 4, A = 21 -------->Ce4k = 21 -----(3)

Taking power 4 in (2), we get C4e4k=604 -----(4)

Dividing (4) by (3), we get

C3 = 604/21

Using log, we have C = 85.15 (app.)

Step 7 :

Plugging   C = 85.15 in (1), we get

A = 85.15 ekt

Our aim is to find the initial weight of the substance.

That is,

the value of "A" when t = 0. Because "A" stands for untransformed substance and  t = 0 is the initial time.

Therefore, to find the no. of grams of the substance initially, we have to plug t = 0 in (1).

When we do so, we get

A = 85.15 ek(0)

A = 85.15 e0

A = 85.15 (1)

A = 85.15

Hence initially there was 85.15 gms (approximately) of the substance.

(Important Note: In this problem, we have not found the value of "k". Because, when we find the initial weight of the substance, we plug t = 0. When t = 0, "k" vanished and the value of "k" is not needed.)

When students look at the steps of the above problem, we hope that the students would have received answer for the question "How to solve differential equation word problems ?".

## Shortest way to solve the above problem Please click the below links to know "How to solve word problems in each of the given topics"

1. Solving Word Problems on Simple Equations

2. Solving Word Problems on Simultaneous Equations

3. Solving Word Problems on Quadratic Equations

4. Solving Word Problems on Permutations and Combinations

5. Solving Word Problems on HCF and LCM

7. Solving Word Problems on Time and Work

8. Solving Word Problems on Trains

9. Solving Word Problems on Time and Work.

10. Solving Word Problems on Ages.

11.Solving Word Problems on Ratio and Proportion

12.Solving Word Problems on Allegation and Mixtures.

13. Solving Word Problems on Percentage

14. Solving Word Problems on Profit and Loss

15. Solving Word Problems Partnership

16. Solving Word Problems on Simple Interest

17. Solving Word Problems on Compound Interest

18. Solving Word Problems on Calendar

19. Solving Word Problems on Clock

20. Solving Word Problems on Pipes and Cisterns

21. Solving Word Problems on Modular Arithmetic

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