State True or False for the following:

(i) The integrating factor of a differential equation of the form $\frac{dy}{dx}+P_{1}x=Q_1$ is given by $e^{\int P_{1}dy}.$

(ii) solution of a differential equation of the type $\frac{dy}{dx}+P_{1}x=Q_1$ is given by $y\cdot I.F.=\left(I.F.\right)Q_{1}dy.$

(iii) Correct substitution for solving the differential equation of the type $\frac{dy}{dx}=f\left(x,y\right),$ where $f\left(x,y\right)$ is a homogeneous function of zero degree, is $y=vx$ .

(iv) Correct substitution for the solution of the differential equation of the type dx/dy = g(x, y) where g (x, y) is a homogeneous function of the degree zero is x = vy.

(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two.

(vi) The differential equation representing the family of circles x² + (y – a)² = a² will be of order two.

(vii) The solution of dy/dx = (y/x)^1/3 is y ^2/3-x^2/3 = c.

(viii) Differential equation representing the family of curves

 y e^x (A cos x + B sin x is d² y/dx² – 2dy/dx + 2y = 0

(ix) The solution of the differential equation dy/dx = x + 2y/x is x + y = kx².

 (x) Solution of xdy/dx = y +x tan y/x is sin (y/x) =cx        

(xi) The differential equation of all non- horizontal lines in a plane is d² y/dx²=0

 

It's difficult but in some colleges you may can get 

  $\overrightarrow{a}+5\overrightarrow{b}$  is collinear with $\overrightarrow{c}$  

⇒   $\overrightarrow{a}+5\overrightarrow{b}$ = $\overrightarrow{c}$           …(1)

$\overrightarrow{b}+6\overrightarrow{c}$ is collinear with $\overrightarrow{a}$  

⇒   $\overrightarrow{b}+6\overrightarrow{c}=\mu \overrightarrow{a}$               …(2)

From (1) and (2)

  $\overrightarrow{b}+6\overrightarrow{c}=\mu \left(\lambda \overrightarrow{c}-5\overrightarrow{b}\right)$          

-> $\left(1+5\mu \right)\overrightarrow{b}+\left(6-\lambda \mu \right)\overrightarrow{c}=0$

$?\overrightarrow{b}$ and $\overrightarrow{c}$

${\displaystyle \int \frac{\left(sinx-cosx\right)si{n}^{2}x}{tanx\left(si{n}^{3}x+co{s}^{3}x\right)}dx}$

${\displaystyle \int \frac{\left(sinx-cosx\right)sinxcosx}{si{n}^{3}x+co{s}^{3}x}dx}$ , put sin³x + cos³x = t(3 sin²x×cosx – 3cos²xsinx) dx = dt

-> $\frac{1}{3}{\displaystyle \int \frac{dt}{t}}$

$=\frac{lnt}{3}+c$

$=\frac{ln\left|si{n}^{3}x+co{s}^{3}x\right|}{3}+c$

$f\left(x\right)=\frac{\left(2^x+2^{-x}\right)tanx\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}}{{\left(7x^2-3x+1\right)}^3}$

$f\left(x\right)=\left(2^x+2^{-x}\right).tanx.\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}.{\left(7x^2-3x+1\right)}^{-3}$

$f\text{'}\left(x\right)=\left(2^x+2^{-x}\right).se{c}^{2}x.\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}.{\left(7x^2-3x+1\right)}^{-3}+tanx.\left(Q\left(x\right)\right)$

$f\text{'}\left(0\right)=2.1\sqrt[]{\frac{\pi }{4}}.1$

$=\sqrt[]{\pi }$