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Quiz-4 / TUT0302

« on: October 18, 2019, 02:12:38 PM »

Find General Solution:
$y''+2y'+y=2e^{-t}$


$r^2+2r+r=0$
$(r+1)(r+1)=0$
$r=-1,-1$

$y=c_{1}e^{-t}+c_2te^{-t}$
$y_{p}(t)=Ae^{-t} \Rightarrow Ate^{-t} \Rightarrow At^2e^{-t}$
$            =$

$y_{p}'(t)=A2te^{-t}+At^2e^{-t}$
$y_{p}''(t)=2Ae^{-t}+2Ate^{-t}+2At-e^{-t}+At^2e^{-t}$
$y_{p}''(t)=2Ae^{-t}-e^t2At-e^{-t}2At+At^2e^{-t}$
$y_{p}''(t)=2Ae^{-t}-4Ae^{-t}+At^2e^{-t}$

$Y=2Ae^{-t}-4Ate^{-t}+At^2e^{-t}+2(A2te^{-t}+At^2(-1)(e^{-t}))+At^2e^{-t}$
$Y=2Ae^{-t}-4Ate^{-t}+At^2e^{-t}+4Ate^{-t}-2At^2e^{-t}+At^2e^{-t}=2e^{-t}$

$=2Ae^{-t}=2e^{-t}$
$A=1$

$Y=C_1e^{-t}+C_2te^{-t}+t^2e^{-t}$

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Quiz-3 / TUT0302

« on: October 11, 2019, 02:00:15 PM »

Find the Wronskian of two solutions of the given differential equation.
$x^2y''+xy'+(x^2-v^2)y=0$

$y''+\frac{x}{x^2}y'+\frac{x^2-v^2}{x^2}y=0$
$y''+\frac{1}{x}y'+{x^2+y^2}{x^2}y=0$

$p(t)=\frac{1}{x}$
$=cexp(-\int \frac{1}{x}dx)$
$=cexp(-lnx)$
$=ce^{-lnx}$
$=c\frac{1}{x}$

$W=\frac{c}{x}$

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Quiz-2 / TUT0302

« on: October 04, 2019, 02:01:09 PM »

Problem: Show that the given equation becomes exact when multiplied by the integrating factor and solve the equation.
$x^{2}y^{3}+x(1+y^{2})y'=0$

$My=x^{2}3y^{2} \neq Nx=\frac{1}{2}x^{2}$
Therefore it is not exact.

Multiply $\mu(\frac{1}{xy^{3}})$ to both side:
$\frac{x^{2}y^{3}}{xy^{3}}+(\frac{x+xy^{2}}{xy^{3}})y'=0$
$=x+(\frac{x}{xy^{3}}+\frac{xy^{2}}{xy^{3}})y'=0$
$=x+(\frac{1}{y^{3}}+\frac{1}{y})y'=0$
$M=x$   
$N=\frac{1}{y^{3}}+\frac{1}{y}$
$My=0 = Nx=0$
Now it is exact.

Integrate M with respect to $x$ we get:
$\phi(x,y)=\frac{1}{2}x^{2}+h(y)$
Take derivative with respect to $y$ on both side:
$\phi y=N=0+h'(y)$
So $h'(y)=(\frac{1}{y^{3}}+\frac{1}{y})$
$h(y)=\int y^{-3}+y^{-1}dy$
$h(y)=-\frac{1}{4}y^{-4}+lny+C$

General Solution: $\frac{1}{2}x^{2}-\frac{1}{4}y^{-4}+lny=C$