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Solve the following differential equation;$(D^{2}-1)$$y=\cos$$2x-2\sin$$2x$

1 Answer

  • A general second-order homogeneous equation is of the form $a\large\frac{d^2y}{dx^2}$$+b\large\frac{dy}{dx}$$+cy=X$
  • Where $X$ is a function of $x$
  • The solution is obtained in two parts.
  • The first part is the complementary function CF
  • This is obtained by solving the equation $am^2+bm+c=0$.The second part is called the particular integral or PI
  • The GS is $y=CF+PI$
  • Let $m_1,m_2$ be the roots of the CE
  • Case 1: $m_1,m_2$ are real numbers and distinct
  • $CF=Ae^{m_1x}+Be^{m_2x}$
  • Case 2: $m_1,m_2$ are complex (i.e)$m_1=\alpha+i\beta$ and $m_2=\alpha-i\beta$
  • Then $CF=e^{\alpha x}[A\cos\beta x+B\sin \beta x]$
  • Case 3: $m_1,m_2$ are real and equal say $m_1$
  • PI :When $x$ is of the form $\cos\alpha x$ or $\sin\alpha x$
  • Case 1 : When $f$ is a function of $D^2$
  • PI =$\large\frac{1}{f(D)} $$\cos\alpha x(or \sin\alpha x)=\large\frac{1}{\phi(D^2)}$$\cos\alpha x=\large\frac{1}{\phi(-\alpha ^2)}$$\cos\alpha x$
  • Case 2:When $f=\phi(D,D^2)$
  • PI is obtained by replacing $D^2$ by $-\alpha^2$
  • $PI=\large\frac{1}{-\alpha ^2+bD+1}$$\cos\alpha x(or \sin\alpha x)$
  • $\;\;\;=\large\frac{1}{bD+P}$$\cos\alpha x(or \sin\alpha x)$
  • $\;\;\;=\large\frac{bD-P}{(bD+P)(bD-P)}$$\cos\alpha x(or \sin\alpha x)$
  • $\;\;\;=\large\frac{bD-P}{(bD^2-P^2)}$$\cos\alpha x(or \sin\alpha x)$
  • $\;\;\;=\large\frac{bD-P}{(b(-\alpha^2)-P^2)}$$\cos\alpha x(or \sin\alpha x)$
  • The denominator is a constant and the numerator represents a differentiation
  • Case 3: If $\phi(-\alpha^2)=0$ then
  • PI=$\large\frac{1}{\phi(D^2)}$$\cos \alpha x=\large\frac{1}{D^2+\alpha^2}$$\cos\alpha x$
  • $\Rightarrow$ R.P of $\large\frac{1}{(D+i\alpha)(D-i\alpha)}$$e^{i\alpha x}$
  • $\Rightarrow$ R.P of $\large\frac{1}{\theta(i\alpha)}$$e^{i\alpha x}$
  • $\Rightarrow$ $\large\frac{-x}{2\alpha}$$(-\sin \alpha x)=\large\frac{x\sin\alpha x}{2\alpha}$
  • or $PI=\large\frac{1}{\phi(D^2)}$$\sin\alpha x$=I.P of $\large\frac{1}{(D+i\alpha)(D-i\alpha)}$$e^{i\alpha x}$
  • PI=$\large\frac{-x}{2\alpha}$$\cos\alpha x$
Step 1:
CE : $m^2-1=0$
Step 2:
PI =$\large\frac{1}{D^2-1}$$\cos 2x-\large\frac{1}{D^2-1}$$2\sin 2x$
Step 3:
$PI_1=\large\frac{1}{D^2-1}$$\cos 2x=\large\frac{1}{-4-1}$$\cos 2x=\large\frac{-1}{5}$$\cos 2x$
Step 4:
$PI_2=\large\frac{-2}{D^2-1}$$\sin 2x=\large\frac{-2\sin 2x}{-4-1}$$=\large\frac{2}{5}$$\sin 2x$
Step 5:
GS : $y=Ae^{-x}+Be^x-\large\frac{1}{5}$$\cos 2x+\large\frac{2}{5}$$\sin 2x$
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