Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 8 Differential Equations to help strengthen their preparations for exams.

## TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Very Short Answer Type Questions

Question 1.

Find the order and degree of \(\frac{d y}{d x}=\frac{x^{1 / 2}}{y^{1 / 2}\left(1+x^{1 / 2}\right)}\)

Solution:

Order is 1 and Degree is ‘1’

Since there is first order derivative with highest degree is ‘1’.

Question 2.

Find the degree and order of the differential equation \(\frac{d^2 y}{d x^2}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{5 / 3}\)

Solution:

The equation can be written as \(\left(\frac{d^2 y}{d x^2}\right)^3=\left[1+\left(\frac{d y}{d x}\right)^2\right]^5\)

The order is 2 and degree is ‘3’

Question 3.

Find the order and degree of the equation

\(1+\left(\frac{d^2 y}{d x^2}\right)^2=\left[2+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}\)

Solution:

The equation can be expressible as

\(\left[1+\left(\frac{d^2 y}{d x^2}\right)^2\right]^2=\left[2+\left(\frac{d y}{d x}\right)^2\right]^3\)

Order is 2 and degree is 4.

Question 4.

Find the order and degree of \(\frac{d^2 y}{d x^2}+2 \frac{d y}{d x}+y=\log \left(\frac{d y}{d x}\right)\)

Solution:

Order is 2and degree is not defined since the equation cannot be expressed as a polynomial equation In the derivatives.

Question 5.

Find the order and degree of \(\left[\left(\frac{d y}{d x}\right)^{\frac{1}{2}}+\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{3}}\right]^{\frac{1}{4}}=0\)

Solution:

Question 6.

Find the order and degree of = \(\frac{d^2 y}{d x^2}=-p^2 y\)

Solution:

Equation is a polynomial equation in \(\frac{d^2 y}{d x^2}\)

So degree is ‘1′ and order is ‘2’.

Question 7.

Find the order and degree of \(\left(\frac{d^3 y}{d x^3}\right)^2-3\left(\frac{d y}{d x}\right)^2-e^x=4\)

\(\left(\frac{d^3 y}{d x^3}\right)^2-3\left(\frac{d y}{d x}\right)^2-e^x=4\)

Solution:

The equation is a polynomial equation in and \(\frac{d y}{d x}\) \(\frac{\mathrm{d}^3 \mathrm{y}}{\mathrm{dx}^3}\)

∴ Order is 3 and degree is 2.

Question 8.

Find the order and degree of \(x^{\frac{1}{2}}\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{3}}+x \frac{d y}{d x}+y=0\)

Solution:

Question 9.

Find the order and degree of \(\left[\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3\right]^{\frac{6}{5}}=6 y\)

Solution:

The given equation can be written as

\(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3=(6 y)^{5 / 6}\)

order is ‘2’ and degree is ‘1’.

Question 10.

Find the order of the differential equation corresponding to y = Ae^{x} + Be^{3x} + Ce^{5x} (A, B, C are parameters) is a solution.

Solution:

Since there are 3 constants in

y = Ae^{x} + Be^{3x} + Ce^{5x} we can have a differential equation of third order by eliminating A,B,C.

∴ Order of the differential equation is ‘3’.

Question 11.

Form the differential equation to y = cx – 2c^{2} where c is a parameter.

Solution:

Given y = cx-2c^{2} ………….. (1)

we have y_{1}=c ……………….. (2)

∴From(1)

y=xy_{1 }– 2y^{2}_{1} ………………….. (3)

∴ This is a differential equation corresponding to (1).

Question 12.

Form the differential equation corresponding to y = A cos 3x+ B sin 3x where A and B are parameters.

Solution:

Question 13.

Express the following differential equations in the form f(x) dx + g(y) dy = 0

(i) \( \frac{d y}{d x}=\frac{1+y^2}{1+x^2}\)

Solution:

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

(ii) \(y-x \frac{d y}{d x}=a\left(y^2+\frac{d y}{d x}\right)\)

Solution:

(iii) \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)

Solution:

(iv) \(\frac{d y}{d x}+x^2=x^2 e^{3 y}\)

Solution:

Question 14.

Find the general solution of x + y \(\frac{dy}{dx}\) = 0.

Solution:

The given equation can be written as

x dx + y dy = 0

∴ ∫ xdx+∫ ydy = c

⇒ x^{2} + y^{2} = 2c

Question 15.

Find the general solution of \(\frac{d y}{d x}=e^{x+y}\)

Solution:

The given equation can be written as \(\frac{d y}{d x}=e^x \cdot e^y\)

writing in variable separable form e^{x }dx = e^{-y }dy = 0

∴ e^{x} + e^{-y} = c is the required solution.

Question 16.

Find the degree of the following homogeneous functions.

(i) f(x, y) = 4x^{2}y + 2xy^{2}

Solution:

Given f(x, y) = 4x^{2}y+2xy^{2
}we have f(kx, ky) = 4k^{2}x^{2}ky + 2kxk^{2}y^{2}

⇒ 4k^{3}x^{2}y + 2k^{3}xy^{2}

⇒ k^{3}(4x^{2}y + y^{2})

⇒ k^{3} f(x, y) ∀ k

and f(x, y), x^{3} Φ \(\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\) and hence f(x, y) is a homogeneous function of degree ‘3’.

(ii) g(x,y)=xy^{1/2}+yx^{1/2}

Solution:

Given g(x, y) =xy^{1/2}+ yx^{1/2}

g(kx, ky) = kx(ky)^{1/2} + (ky)(kx)^{1/2}

⇒ k^{3/2} (xyk^{1/2} + yx^{1/2})

⇒ k^{3/2} g(x, y)

∴ g(x, y) is a homogeneous function of degree ‘3’.

(iii) \(h(x, y)=\frac{x^2+y^2}{x^3+y^3}\)

Solution:

∴ h(x, y) is a homogeneous function of degree – 1.

(iv) Show that f(xy) = I +e^{x/y }is a homogeneous function of x and y.

Solution:

(v) f(x,y) = x \(\sqrt{\mathbf{x}^2+y^2}-y^2\) is a homogeneous function of x and y.

Solution:

∴f(x, y) is a homogeneous function of degree ‘1’.

(vi) f(x,y) = x – y log y + y log x

Solution:

Givenf(x, y) =x-ylogy+ylogx

∴ f(kx, ky) – kx – ky log (ky) + ky log(kx)

= k[x-y log(ky) + ylog(kx)]

= k[x- y(logk+logy) +y(logk+logx)]

= k[x – y log y + y log x]

= k f(x, y)

∴ f(x, y) is a homogeneous function of degree ‘F.

Question 17.

Express (1+e^{x/y}) dx + e^{x/y }\(\left(1-\frac{x}{y}\right)\) dy = 0 in the form \(\frac{\mathbf{d x}}{\mathbf{d y}}=F\left(\frac{x}{y}\right)\)

Solution:

Question 18.

Express \(\left(x \sqrt{x^2+y^2}-y^2\right)\) dx+xy dx = 0 in the form \(\frac{\mathbf{d y}}{\mathbf{d x}}=F\left(\frac{x}{y}\right)\)

Solution:

Question 19.

Express \(\frac{d y}{d x}=\frac{y}{x+y e^{-\frac{2 x}{y}}}\) in the form \(\frac{d x}{d y}=F\left(\frac{x}{y}\right)\)

Solution:

Question 20.

Transform x logx \(\frac{d y}{d x}\) y into linear form.

Solution:

Dividing both sides by x log x we get

Question 21.

Transform \(\left(x+2 y^3\right) \frac{d y}{d x}=y\) into linear form

Solution:

Question 22.

Find I.F. of the following differential equations by converting them into linear form.

(i) cosx\(\frac{d y}{d x}\)+y sinx=tanx

Solution:

(ii) (2y -10y^{3}) \(\frac{d y}{d x}\) + y = 0

Solution:

Short Answer Type Questions

Question 1.

Find the order of the differential equation corresponding to y = c( x- c)^{2} where c is an arbitrary constant

Solution:

Given y = c(x – e)^{2}; eliminate ‘c’ and form the differential equation.

Question 2.

Form the differential equation corresponding to the family of circles of radius ‘r’ given by (x-a)^{2}+(y-b)^{2}=r^{2 }where a and b are parameters.

Solution:

Question 3.

Form the differential equation corresponding to the family of circles passing through the origin and having centres on Y- axis.

Solution:

The equation of family of circles passing through the origin and having centres on Y-axis is

x^{2}+y^{2}-2fy=0 ……………….. (1)

Differentiating w.r.t x, we get

Question 4.

Solve \(y^2-x \frac{d y}{d x}=a\left(y+\frac{d y}{d x}\right)\)

Solution:

Question 5.

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

Solution:

The equation can be written as

Question 6.

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

Solution:

Question 7.

Find the equation of the curve whose slope at any point (x, y) is \(\frac{y}{x^2}\) and which satisfy the condition y = 1 when x =3.

Solution:

We have the slope at any point x, y) on the

Question 8.

Solve y (1+x)dx+x(1+y) dy = 0

Solution:

The given equation can be expressed as

logx+x+logy+y=c

x + y + log (xy) = c which is the required solution.

Question 9.

Solve \(\frac{d y}{d x}\) = sin(x + y) +cos(x + y)

Solution:

Question 10.

Solve that (x – y)^{2 } \(\frac{d y}{d x}=a^2\)

Solution:

Question 11.

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

Solution:

Question 12.

Solve \(\frac{d y}{d x}=\sqrt{y-x}\)

Solution:

Question 13.

Solve \(\frac{d y}{d x}\) +1 = e^{x+y}

Solution:

Question 14.

Solve \(\frac{d y}{d x}\) = (3x + y + 4)^{2
}Solution:

Question 15.

Solve \(\frac{d y}{d x}\) – x tan(y-x)= 1

Solution:

Question 16.

Solve \(\frac{d y}{d x}=\frac{y^2-2 x y}{x^2-x y}\)

Solution:

The given equation ¡s a homogeneous equation of degree ‘2’.

which is athe general solution of the given equation.

Question 17.

Solve(x^{2}+y^{2})dx=Zxydy

Solution:

The given equation can be written as

Question 18.

Solve xy^{2}dy – (x^{3}+y)dx=0

Solution:

The given equation can be written as \(\frac{d y}{d x}=\frac{x^3+y^3}{x y^2}\) which is a homogeneous equation.

which is the general solution of the given equation.

Question 19.

Solve \(\frac{d y}{d x}=\frac{x^2+y^2}{2 x^2}\)

Solution:

The given equation \(\frac{d y}{d x}=\frac{x^2+y^2}{2 x^2}\) homogeneous equation.

which is the general solution of the given equation.

Question 20.

Give the solution of x sin^{2} \(\left(\frac{y}{x}\right)\) dx = y dx – x dy which passes through the point \(\left(1, \frac{\pi}{4}\right)\)

Solution:

is the required particular solution of the given equation.

Question 21.

Solve(x^{3}-3xy^{2})dx+(3x^{2}y-y^{3})dy=0

Solution:

The given equation can be written as

Question 22.

Solve the equation \(\frac{d y}{d x}=\frac{3 x-y+7}{x-7 y-3}\)

Solution:

Here a=3, b =-1,c = 7

a’=1, b’=-7,c’ = -3

and b =- a’. Hence that solution can be obtained by grouping.

∴ From the given equation

3xdx – ydx+7dx = xdy-7ydy – 3dy

= (xdy+ydx) – 7ydy – 7dx – 3xdx – 3dy = 0

= ∫d(xy) -∫7ydy – 7∫dx – 3∫xdx – 3∫dy = 0

= xy – 7\(\frac{y^2}{2}\) – 7x -3 \(\frac{x^2}{2}\) -3y =c

⇒ 2xy – 7y^{2}-14x-3x^{2}– 6y=2c

⇒ 2xy – 7y^{2} – 14x-3x^{2} – 6y= c’ where C – 2c

Is the required solution.

Question 23.

Solve (1+x^{2}) \(\frac{\mathrm{dy}}{\mathbf{d x}}\) +2xy = 4x^{2
}Solution:

Question 24.

Solve sin ^{2 }x \(\frac{d y}{d x}\) +y = cot x

Solution:

Question 25.

Find the solution of the equation x(x – 2) \(\frac{d y}{d x}\) (x – 1)y=x^{3}(x-2) which sotisfies the condition that y=9 where x=3.

Solution:

The equation can be written as

Question 26.

Solve (1+y^{2})dx = (tan^{-1} y-x)dy

Solution:

The given equation can be written as

Long Answer Type Questions

Question 1.

Solve \(\sqrt{1+x^2} \sqrt{1+y^2}\)dx + xy dy =0.

Solution:

The given equation can he written as

Is the solution of the given differential equation.

Question 2.

Solve x sec \(\left(\frac{\mathbf{y}}{\mathbf{x}}\right)\) (y dx+xdy)=y cosec \(\left(\frac{\mathbf{y}}{\mathbf{x}}\right)\)

Solution:

The given equation can be written as

which is the general solution of the given equation.

Question 3.

Solve (2x+y+3)dx=(2y+x+1)dy

Solution: