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Differential equations - Notes, Concept and All Important Formula

DIFFERENTIAL EQUATION

1. DIFFERENTIAL EQUATION:

An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a DIFFERENTIAL EQUATION.


2. SOLUTION (PRIMITIVE) OF DIFFERENTIAL EQUATION :

Finding the unknown function which satisfies given differential equation is called SOLVING OR INTEGRATING the differential equation. The solution of the differential equation is also called its PRIMITIVE, because the differential equation can be regarded as a relation derived from it.


3. ORDER OF DIFFERENTIAL EQUATION:

The order of a differential equation is the order of the highest order differential coefficient occurring in it.


4. DEGREE OF DIFFERENTIAL EQUATION:

The degree of a differential equation which can be written as a polynomial in the derivatives, is the degree of the derivative of the highest order occurring in it, after it has been expressed in a form free from radicals & fractions so far as derivatives are concerned, thus the differential equation:

\(\mathrm{f}(\mathrm{x}, \mathrm{y})\left[\dfrac{\mathrm{d}^{\mathrm{m}} \mathrm{y}}{\mathrm{dx}^{\mathrm{m}}}\right]^{\mathrm{p}}\)\(+\phi(\mathrm{x}, \mathrm{y})\left[\dfrac{\mathrm{d}^{\mathrm{m}-1}(\mathrm{y})}{\mathrm{dx}^{\mathrm{m}-1}}\right]^{\mathrm{q}}+\ldots\)\( \ldots \ldots=0\) is of order \(\mathrm{m}\) & degree \(\mathrm{p}\).

Note that in the differential equation \(e^{\mathrm{y'''}}-\mathrm{xy}^{\prime \prime}+\mathrm{y}=0\) order is three but degree doesn't exist.


5. FORMATION OF A DIFFERENTIAL EQUATION:

If an equation in independent and dependent variables having some arbitrary constant is given, then a differential equation is obtained as follows:

(a) Differentiate the given equation w.r.t the independent variable (say \(\mathrm{x}\) ) as many times as the number of independent arbitrary constants in it.

(b) Eliminate the arbitrary constants.

The eliminant is the required differential equation.

Note : A differential equation represents a family of curves all satisfying some common properties. This can be considered as the geometrical interpretation of the differential equation.


6. GENERAL AND PARTICULAR SOLUTIONS :

The solution of a differential equation which contains a number of independent arbitrary constants equal to the order of the differential equation is called the GENERAL SOLUTION (OR COMPLETE INTEGRAL OR COMPLETE PRIMITIVE). A solution obtainable from the general solution by giving particular or initial values to the constants is called a PARTICULAR SOLUTION.


7. ELEMENTARY TYPES OF FIRST ORDER & FIRST DEGREE DIFFERENTIAL EQUATIONS :

(a) Variables separable :

TYPE-1 : If the differential equation can be expressed as ; \(\mathrm{f}(\mathrm{x}) \mathrm{dx}+\mathrm{g}(\mathrm{y}) \mathrm{dy}=0\) then this is said to be variable \(-\) separable type. A general solution of this is given by \(\displaystyle \int f(x) d x+\displaystyle \int g(y) d y=c\); where \(c\) is the arbitrary constant. Consider the example \((\mathrm{dy} / \mathrm{dx})\) \(=e^{x-y}+x^{2} \cdot e^{-y} .\)

TYPE-2 : Sometimes transformation to the polar co-ordinates facilitates separation of variables. In this connection it is convenient to remember the following differentials. If \(\mathrm{x}=\mathrm{r} \cos\) \(\theta, y=\operatorname{rsin} \theta\) then,

(i) \(x d x+y d y=r d r\)

(ii) \(\mathrm{dx}^{2}+\mathrm{dy}^{2}=\mathrm{dr}^{2}+\mathrm{r}^{2} \mathrm{~d} \theta^{2}\)

(iii) \(x d y-y d x=r^{2} d \theta\)

Also, if \(x=r \sec \theta \) & \(y=r \tan \theta\) then

\(x d x-y d y=r d r\) and \(x d y-y d x=r^{2} \sec \theta d \theta\)

TYPE-3: \(\dfrac{d y}{d x}=f(a x+b y+c), b \neq 0\)

To solve this, substitute \(\mathrm{t}=\mathrm{ax}+\mathrm{by}+\mathrm{c}\). Then the equation reduces to separable type in the variable \(\mathrm{t}\) and \(\mathrm{x}\) which can be solved.

Consider the example \((\mathrm{x}+\mathrm{y})^{2} \dfrac{\mathrm{d} y}{\mathrm{dx}}=\mathrm{a}^{2}\)

(b) Homogeneous equations:

A differential equation of the form \(\dfrac{\mathrm{dy}}{\mathrm{dx}}=\dfrac{\mathrm{f}(\mathrm{x}, \mathrm{y})}{\phi(\mathrm{x}, \mathrm{y})}\), where \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) & \(\phi(\mathrm{x}, \mathrm{y})\) are homogeneous functions of \(\mathrm{x} \) & \( \mathrm{y}\) and of the same degree, is called HOMOGENEOUS. This equation may also be reduced to the form \(\dfrac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{g}\left(\dfrac{\mathrm{x}}{\mathrm{y}}\right) \) & is solved by putting \(\mathrm{y}=\mathrm{vx}\) so that the dependent variable \(y\) is changed to another variable \(\mathrm{v}\), where \(\mathrm{v}\) is some unknown function. The differential equation thus, is transformed to an equation which is variables separable.

Consider the example \(\dfrac{d y}{d x}+\dfrac{y(x+y)}{x^{2}}=0\)

(c) Equations reducible to the homogeneous form:

If \(\dfrac{d y}{d x}=\dfrac{a_{1} x+b_{1} y+c_{1}}{a_{2} x+b_{2} y+c_{2}} ;\) where \(a_{1} b_{2}-a_{2} b_{1} \neq 0\), i.e. \(\dfrac{a_{1}}{b_{1}} \neq \dfrac{a_{2}}{b_{2}}\) then the substitution \(\mathrm{x}=\mathrm{u}+\mathrm{h}, \mathrm{y}=\mathrm{v}+\mathrm{k}\) transform this equation to a homogeneous type in the new variables \(\mathrm{u}\) and \(\mathrm{v}\) where \(\mathrm{h}\) and \(\mathrm{k}\) are arbitrary constants to be chosen so as to make the given equation homogeneous.

Note:

(i) If \(\mathrm{a}_{1} \mathrm{~b}_{2}-\mathrm{a}_{2} \mathrm{~b}_{1}=0\), then a substitution \(\mathrm{u}=\mathrm{a}_{1} \mathrm{x}+\mathrm{b}_{1} \mathrm{y}\) transforms the differential equation to an equation with variables separable.

(ii) If \(\mathrm{b}_{1}+\mathrm{a}_{2}=0\), then a simple cross multiplication and substituting \(\mathrm{d}(\mathrm{xy})\) for \(\mathrm{xdy}+\mathrm{ydx}\) & integrating term by term yields the result easily.

Consider the examples \(\dfrac{d y}{d x}=\dfrac{x-2 y+5}{2 x+y-1} ; \dfrac{d y}{d x}=\dfrac{2 x+3 y-1}{4 x+6 y-5}\) & \( \dfrac{d y}{d x}=\dfrac{2 x-y+1}{6 x-5 y+4}\)

(iii) In an equation of the form: \(y f(x y) d x+x g(x y) d y=0\) the variables can be separated by the substitution \(\mathrm{xy}=\mathrm{v}\).


8. LINEAR DIFFERENTIAL EQUATIONS :

A differential equation is said to be linear if the dependent variable & its differential coefficients occur in the first degree only and are not multiplied together.

The nth order linear differential equation is of the form;

\(a_{0}(x) \dfrac{d^{n} y}{d x^{n}}\)\(+a_{1}(x) \dfrac{d^{n-1} y}{d x^{n-1}}\)\(+\ldots\)\( ..+a_{n-1}(x) \cdot y\)\(+a_{n}(x)=0\)

where \(\mathrm{a}_{0}(\mathrm{x}), \mathrm{a}_{1}(\mathrm{x}), \ldots, \mathrm{a}_{\mathrm{n}}(\mathrm{x})\) are called the coefficients of the differential equation.

(a) Linear differential equations of first order:

The most general form of a linear differential equations of first order is \(\dfrac{d y}{d x}+P y=Q\), where \(P \) & \( Q\) are functions of \(x\). To solve such an equation multiply both sides by integrating factor \(e^{^{ \int \operatorname{Pdx}}}\).

Then the solution of this equation will be \(y e^{^{ \int \mathrm{Pdx}}}=\displaystyle \int \mathrm{Qe}^{^{ \int \mathrm{Pdx}}} \mathrm{dx}+\mathrm{c}\)

(b) Equations reducible to linear form:

The equation \(\dfrac{\mathrm{d} y}{\mathrm{dx}}+\mathrm{Py}=\mathrm{Q} \cdot \mathrm{y}^{\mathrm{n}}\) where \(\mathrm{P} \) & \(\mathrm{Q}\) are function, of \(\mathrm{x}\) is reducible to the linear form by dividing it by \(y^{n} \) & then substituting \(y^{n+1}=\) Z. Consider the example \(\left(x^{3} y^{2}+x y\right) d x=d y\). The equation \(\dfrac{\mathrm{d} y}{\mathrm{dx}}+\mathrm{Py}=\mathrm{Qy}^{n}\) is called BERNOULI'S EQUATION.


9. TRAJECTORIES :

A curve which cuts every member of a given family of curves according to a given law is called a Trajectory of the given family.

Orthogonal trajectories:

A curve making at each of its points a right angle with the curve of the family passing through that point is called an orthogonal trajectory of that family.

We set up the differential equation of the given family of curves. Let it be of the form \(\mathrm{F}\left(\mathrm{x}, \mathrm{y}, \mathrm{y}^{\prime}\right)=0\)

The differential equation of the orthogonal trajectories is of the form \(F\left(x, y, \dfrac{-1}{y^{\prime}}\right)=0\)

The general integral of this equation \(\phi(x, y, C)=0\) gives the family of orthogonal trajectories.

Note:

Following exact differentials must be remembered :

(i) \( d x+d y=d(x+y)\)

(ii) \(d x-d y=d(x-y)\)

(iii) \(x d y+y d x=d(x y)\)

(iv) \(\dfrac{x d y-y d x}{x^{2}}=d\left(\dfrac{y}{x}\right)\)

(v) \(\dfrac{y d x-x d y}{y^{2}}=d\left(\dfrac{x}{y}\right)\)

(vi) \(2(x d x+y d y)=d\left(x^{2}+y^{2}\right)\)

It should be observed that :

(i) \(\dfrac{x d y+y d x}{x y}=d(\ell n x y)\)

(ii) \(\dfrac{d x+d y}{x+y}=d(\ln (x+y))\)

(iii) \(\dfrac{x d y-y d x}{x y}=d\left(\ln \dfrac{y}{x}\right)\)

(iv) \(\dfrac{y d x-x d y}{x y}=d\left(\ln \dfrac{x}{y}\right)\)

(v) \(\dfrac{x d y-y d x}{x^{2}+y^{2}}=d\left(\tan ^{-1} \dfrac{y}{x}\right)\)

(vi) \(\dfrac{y d x-x d y}{x^{2}+y^{2}}=d\left(\tan ^{-1} \dfrac{x}{y}\right)\)

(vii) \(\dfrac{x d x+y d y}{x^{2}+y^{2}}=d\left[\ln \sqrt{x^{2}+y^{2}}\right]\)

(viii) \(\mathrm{d}\left(-\dfrac{1}{\mathrm{x} y}\right)=\dfrac{\mathrm{xdy}+\mathrm{ydx}}{\mathrm{x}^{2} \mathrm{y}^{2}}\)

(ix) \(\mathrm{d}\left(\dfrac{e^{x}}{y}\right)=\dfrac{y e^{x} d x-e^{x} d y}{y^{2}}\)

(x) \(\mathrm{d}\left(\dfrac{e^{y}}{x}\right)=\dfrac{x e^{y} d y-e^{y} d x}{x^{2}}\)

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