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Differential Equations

Have you ever observed what happens when we deposit our money in bank for a certain period of time?

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The bank gives us interest on the money we have deposited at a certain rate of interest and we get the total amount which is the sum of our original money deposited i.e. the principal & the interest earned for that period of time.

So, if in a bank principal increases continuously at the rate of 5 % per year, we are interested in finding the worth of the money (say 1000 dollars) after 10 years.

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Hence, we require a relation that connects the rate of change of the principal with respect to time i.e. a relation between a dependent variable, independent variable and their rate of change i.e. derivatives.

Here we will introduce the concept of Differential Equations.

 

I. Definition

If y=f(x) be a functionthen x is the independent variable & y is the dependent variable, since its value is dependent on the value/s assigned to x.

In general, an equation which involves independent variable (say x), dependent variable (say y) & the derivatives of the dependent variable with respect to independent variable is called a Differential equation.

Here the derivatives can be of the form  \frac{dy}{dx},\frac{ d^{{2}}y}{dx^{{2}}},\frac{ d^{{3}}y}{dx^{{3}}},…..\frac{ d^{{n}}y}{dx^{{n}}}

 

II. Order of a differential equation.

Order of a differential equation is the order of the highest order derivative present in the differential equation. Here the derivative means the derivative of the dependent variable with respect to the independent variable.

\frac{dy}{dx} means derivative of First order i.e. order 1.

\frac{ d^{{2}}y}{dx^{{2}}} means derivative of Second i.e. order 2.

\frac{ d^{{3}}y}{dx^{{3}}}  means derivative of Third i.e. order 3.

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III. Degree of a differential equation.

Degree of a differential equation is defined only, if the given differential equation is a polynomial equation in its derivatives.

Degree (if defined) is highest power of the highest order derivative occurring i.e. present in the differential equation.

Note 1. For a given differential equation Order and Degree (if defined) are both positive integers i.e. natural numbers.

  1. For identifying the degree of a differential equation, the derivatives must first be made free from radicals, fractions & then expressed as a polynomial equation in its derivatives. Then we identify its degree.

 

EXAMPLES

Now let’s consider some examples on differential equation.

Example 1: Find the order & degree (if defined) of the differential Equation.

\frac{ d^{{3}}y}{dx^{{3}}} +4\left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{3} –\frac{dy}{dx} +2y=0

Here the highest order derivative present in the differential equation is \frac{ d^{{3}}y}{dx^{{3}}} whose order is 3. Hence, its degree is also defined. The highest power of the highest order derivative \left ( \frac{ d^{{3}}y}{dx^{{3}}} \right ) ^{1} is one. Hence the degree is 1.

 

Example 2: Find the Order & degree (if defined) of the differential Equation.

\left ( \frac{ d^{{3}}y}{dx^{{3}}} \right ) ^{4} + \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{3} + 7\left ( \frac{ d^{{}}y}{dx^{{}}} \right ) ^{} + y^{{5}} =0

Here the highest order derivative present in the differential equation is \frac{ d^{{3}}y}{dx^{{3}}}  whose order is 3. Hence the order of the differential equation is 3.

Since, the given differential equation is expressed as a polynomial equation in its derivatives. Hence, its degree is also defined. The highest power of the highest order derivative \left ( \frac{ d^{{3}}y}{dx^{{3}}} \right ) ^{4} is four. Hence the degree is 4.

 

Example 3: Find the Order & degree (if defined) of the differential Equation.

\left ( \frac{ d^{{4}}y}{dx^{{4}}} \right ) ^{2} + \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{3} + Cos\left ( \frac{ d^{{}}y}{dx^{{}}} \right ) ^{}+5=0

Here the highest order derivative present in the differential equation is \left ( \frac{ d^{{4}}y}{dx^{{4}}} \right ) ^{} whose order is 4. Hence the order of the differential equation is 4.

Since, the given differential equation contains the term Cos\left ( \frac{ d^{{}}y}{dx^{{}}} \right ) ^{}, hence it is not a polynomial equation in its derivatives. Hence, its degree is not defined.

 

Example 4: Find the Order & degree (if defined) of the differential Equation.

xy\left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{} – x\left ( \frac{ d^{{}}y}{dx^{{}}} \right ) ^{2} -y = 0

Here the highest order derivative present in the differential equation is \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{}  whose order is 2. Hence the order of the differential equation is 2.

Since, the given differential equation is expressed as a polynomial equation in its derivatives. Hence, its degree is also defined. The highest power of the highest order derivatives \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{1} is one. Hence the degree is 1.

 

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Find the order & degree (if defined) of the given differential equations.

  1. \frac{ d^{{4}}y}{dx^{{4}}} ^{} + 7 \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{3} – \frac{dy}{dx} =0
  2. \left ( \frac{ d^{{4}}y}{dx^{{4}}} \right ) ^{2} + \left ( \frac{ d^{{3}}y}{dx^{{3}}} \right ) ^{3} – 3\left ( \frac{dy}{dx} \right ) + y^{{3}} =0
  3. x\left ( \frac{dy}{dx} \right ) +y=0
  4. \left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{3} + Sin\left ( \frac{dy}{dx} \right ) -12=0
  5. 2x^{{2}}\left ( \frac{ d^{{2}}y}{dx^{{2}}} \right ) ^{}-7\left ( \frac{dy}{dx} \right )+4y=0

Answer Key

  1. Order – 4; Degree – 1
  2. Order- 4; Degree – 2
  3. Order – 1; Degree – 1
  4. Order – 2; Degree – not defined
  5. Order – 2; Degree – 1

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