Diffential Equations-Third Semester Exam Answer Key 2022

திருவண்ணாமலை மாவட்டம் அரசு கலைக் கல்லூரியில் இரண்டாம் ஆண்டு படிக்கும் மாணவர்களுக்கு நடைபெற்றுக் கொண்டிருக்கும் மூன்றாம் பருவ செமஸ்டர் ஆன்லைன் தேர்வுகளுக்கான விடைகள்-வினாக்களுடன் வழங்குகின்றோம்.. மாணவர்கள் இதை படித்து பயன்பெற வேண்டுகிறோம்..

Time : Three hours Maximum

TOTAL – 75 marks

( SECTION -A )

(10 × 2 = 20 marks)

Answer ALL questions.

1. Write the condition for exactness.

Ans :

* A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation.

* The equation P(x, y)y′ + Q(x, y) = 0, or in the equivalent alternate notation.

P(x, y)dy + Q(x, y)dx = 0 ,

is exact if Px(x,y) = Qy(x, y).

2. Solve : y = 2px + ${y^3}{p^3}$

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3. Find the complementary function of $frac{{{d^2}y}}{{d{x^2}}} + 6frac{{dy}}{{dx}} + 5y = {e^{2x}}$

Ans :

4. Find the particular integral of $({D^2} - 2aD + {a^2})y = {e^{ax}}$

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5. Write the general form of simultaneous equations of the first order and first degree.

Ans :

The general form of an equation is ax + by = c. Here, a, and b are non−zero coefficients and c are the constants and x, and y are variables. Two such equations a1x + b1y = c1 and a2x + b2y = c2 are a pair of simultaneous equations in x, and y.

6. State the criterion of integrability of the equation Pdx +Qdy + Rdz = 0 .

Ans :

Consider the total (or single) differential equation

Pdx + Qdy + Rdz = 0 ……..(1)

where P, Q,R are functions of x, y, z.

Let (1.2) have an integral u(x, y, z) = c ……. (2)

Then total differential du must be equal to Pdx + Qdy + Rdz, or to it multiplied by a factor. But,
we know that

$du = left( {frac{{partial u}}{{partial x}}} right)dx + left( {frac{{partial u}}{{partial y}}} right)dy + left( {frac{{partial u}}{{partial z}}} right)dz$ ……… (3)

$lambda P = left( {frac{{partial u}}{{partial x}}} right)$   ,
$lambda Q = left( {frac{{partial u}}{{partial y}}} right)$   ,
$lambda R = left( {frac{{partial u}}{{partial z}}} right)$    …….(4)

$frac{partial }{{partial y}}(lambda P) = frac{{{partial ^2}u}}{{partial ypartial x}}$ = $frac{partial }{{partial x}}(frac{{partial u}}{{partial y}}) = frac{partial }{{partial x}}(lambda Q)$

$lambda frac{{partial P}}{{partial y}} + Pfrac{{partial lambda }}{{partial y}} =$ $lambda frac{{partial lambda }}{{partial x}} + Qfrac{{partial lambda }}{{partial x}}$

$lambda (frac{{partial P}}{{partial y}} - frac{{partial Q}}{{partial x}}) = Qfrac{{partial lambda }}{{partial x}} - Pfrac{{partial lambda }}{{partial y}}$ ……(5)

Similarly, $lambda (frac{{partial Q}}{{partial z}} - frac{{partial R}}{{partial y}}) = Rfrac{{partial lambda }}{{partial y}} - Qfrac{{partial lambda }}{{partial z}}$ …….(6)

and $lambda (frac{{partial R}}{{partial x}} - frac{{partial P}}{{partial z}}) = Pfrac{{partial lambda }}{{partial z}} - Rfrac{{partial lambda }}{{partial x}}$ ……(7)

Multiplying (5)-(6)-(7) by R, P and Q respectively and adding, we get

$P(frac{{partial Q}}{{partial z}} - frac{{partial R}}{{partial y}}) + Q(frac{{partial R}}{{partial x}} - frac{{partial P}}{{partial z}})$ $- R(frac{{partial P}}{{partial y}} - frac{{partial Q}}{{partial x}}) = 0$ ,,

7. Define : Laplace transform.

Ans :

* The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.

The (unilateral) Laplace transform £ (not to be confused with the Lie derivative, also commonly denoted /£) is defined by

$Lf(t) = intlimits_{ - infty }^infty {f(t){e^{ - st}}} dt$

8. Proove : $L(f''(t)) = {s^2}L(f(t)) - sf(0) - f'(0)$

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9. Define : General integral.

Ans :

A general integral of a first-order partial differential equation is a relation between the variables in the equation involving one arbitrary function such that the equation is satisfied when the relation is substituted in it, for every choice of the arbitrary function. See also Integral of a differential equation.

10. Write the Clairaut’s form of the equation.

Ans :

* Clairaut’s equation, in mathematics, a differential equation of the form y = x (dy/dx) + f(dy/dx) where f(dy/dx) is a function of dy/dx only. The equation is named for the 18th-century French mathematician and physicist Alexis-Claude Clairaut, who devised it.

* y(x)=Cx+f(C), the so-called general solution of Clairaut’s equation. y=xy′+(y′).