Solve the differential equation d y d x + y x = x 2 .

Given differential equation is, d y d x + y x = x 2 Which is linear differential equation of first order and is of the form, d y d x + P y = Q Where P = 1 x and Q = x 2 ∴ I . F = e ∫ P d x = e ∫ 1 x d x = e log x = x Multiply given equation by x , we get, x d y d x + y = x 3 ⇒ x y = x 4 4 + c ∴ y = x 3 4 + c x

Solve the differential equation d y d x = x log x

Given d y d x = x log x The above equation can be written as d y = x log x d x Integrate both sides ⇒ ∫ d y = ∫ x log x d x ⇒ y = log x ∫ x d x - ∫ ∫ x d x d d x log x d x (Using integration by part) ⇒ y = x 2 2 log x - ∫ x 2 2 1 x d x ⇒ y = x 2 2 log x - x 2 4 + C This is the required solution for the given differential equation.

Solve the differential equation: x d y - y d x = x 2 + y 2 d x , given that y = 0 when x = 1 .

Given the differential equation is x d y - y d x = x 2 + y 2 d x ⇒ x d y = x 2 + y 2 + y d x ⇒ d y d x = x 2 + y 2 + y x → 1 Here the numerator degree and denominator degree are having the same. It is a homogeneous differential equation Put y = v x and d y d x = v + x d v d x (Apply product rule in differentiation) Now put these values in 1 , we get ⇒ v + x d v d x = x 2 + v 2 x 2 + v x x = 1 + v 2 + v ⇒ x d v d x = 1 + v 2 ⇒ d v 1 + v 2 = d x x ⇒ ∫ d v 1 + v 2 = ∫ d x x (Apply integration) ⇒ log v + 1 + v 2 = log x + log C = log C x ⇒ v + 1 + v 2 = C x ⇒ y x + 1 + y 2 x 2 = C x Put y = v x ⇒ y + x 2 + y 2 = C x 2 → 2 When x = 1 , y = 0 (given), substitute these values in 2 , we get ⇒ 0 + 1 + 0 = C × ( 1 ) 2 ⇒ C = 1 ∴ y + x 2 + y 2 = x 2 is the required solution.

Solve the differential equation 1 + x 2 d y d x + 2 x y - 4 x 2 = 0 , subject to the initial condition y ( 0 ) = 0 .

Given differential equation 1 + x 2 d y d x + 2 x y - 4 x 2 = 0 , y ( 0 ) = 0 ⇒ 1 + x 2 d y d x + 2 x y = 4 x 2 Divide by 1 + x 2 on both sides, we get ⇒ d y d x + 2 x 1 + x 2 · y = 4 x 2 1 + x 2 It is of the form d y d x + P y = Q Here, P = 2 x 1 + x 2 , Q = 4 x 2 1 + x 2 Now, integrating factor I . F = e ∫ P d x ⇒ I . F = e ∫ 2 x 1 + x 2 d x ⇒ I . F = e log 1 + x 2 ∵ ∫ f ' ( x ) f x d x = log f x + C ⇒ I . F = 1 + x 2 ∴ The general solution is y × I . F = ∫ Q × I . F d x + C ⇒ y · 1 + x 2 = ∫ 4 x 2 1 + x 2 × 1 + x 2 d x + C ⇒ y · 1 + x 2 = 4 ∫ x 2 d x + C ⇒ y · 1 + x 2 = 4 x 3 3 + C → 1 Now put x = 0 , y = 0 in 1 , we get ⇒ C = 0 Put the value of C in 1 , we get ⇒ y · 1 + x 2 = 4 x 3 3 .

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Important Questions on Differential Equations

HARD

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Solve

(1 + y^{2}) d x = (\tan^{- 1} y - x) d y

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

\frac{d y}{d x} + \frac{3}{\cos^{2} x} y = \frac{1}{\cos^{2} x}

x \in (- \frac{π}{3}, \frac{π}{3}),

and

y (\frac{π}{4}) = \frac{4}{3},

then

y (- \frac{π}{4})

equals

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Solve the differential equation

\frac{d y}{d x} + \frac{y}{x} = x^{2} .

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Expand VNTR and describe its role in DNA fingerprinting.

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

The solution of

\frac{d y}{d x} + y = e^{- x}, y (0) = 0

is :

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

List any two applications of DNA fingerprinting technique.

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Solve the differential equation

\frac{d y}{d x} = x \log x

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Given below is the representation of the extent of the global diversity of invertebrates. What groups the four portions (A-D) represent respectively?
Question Image

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Find the particular solution of the differential equation

e^{x} \tan y d x + (2 - e^{x}) \sec^{2} y d y = 0

, given that

y = \frac{π}{4}

when

x = 0

HARD

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Solve the differential equation:

x d y - y d x = \sqrt{x^{2} + y^{2}} d x

, given that

y = 0

when

x = 1

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Under what conditions is the phenomenon of total internal reflection of light observed? Obtain the relation between the critical angle of incidence and the refractive index of the medium.

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

(2 + \sin x) \frac{d y}{d x} + (y + 1) \cos x = 0

and

y (0) = 1

, then

y (\frac{π}{2})

is equal to

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

The figure shows a ray of light falling normally on the face AB of an equilateral glass prism having a refractive index $3 / 2$ , placed in water of refractive index $4 / 3$ . Will this ray suffer total internal reflection on striking the face AC? Justify your answer.

Question Image

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Which of the following is not required for any of the techniques of DNA fingerprinting available at present?

EASY

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

The best quality optical fibres are made of :

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

"India has greater ecosystem diversity than Norway." Do you agree with this statement? Give reasons in support of your answer.

HARD

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Write the difference between genetic biodiversity and species biodiversity that exists at all the levels of biological organisations.

HARD

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

Solve the differential equation

(1 + x^{2}) \frac{d y}{d x} + 2 x y - 4 x^{2} = 0

, subject to the initial condition

y (0) = 0

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

A tangent to the curve,

y = f (x)

P (x, y)

meets

x

-axis at

A

and

y

-axis at

B

. If

A P : B P = 1 : 3

and

f (1) = 1,

then the curve also passes through the point

HARD

Mathematics>Integral Calculus>Differential Equations>Methods of Solving First Order, First Degree Differential Equation

y (x)

is the solution of the differential equation

(x + 2) \frac{d y}{d x} = x^{2} + 4 x - 9, x \neq - 2

and

y (0) = 0,

then

y (- 4)

is equal to

Under which redressal agency can a consumer directly approach if the value of the goods or services exceeds Rs. 1 crore?

Important Questions on Differential Equations

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