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Solve the following. $\large\frac{dy}{dx}+\frac{y}{x}$$\;=\;\sin(x^{2})$

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Toolbox:
  • Linear Differential equation.
  • This is of the form $\large\frac{dy}{dx}$$+Py=Q$ where P and Q are functions of x only.
  • The integrating factor $I= e^{\int \large pfd}$ and the $G.S$ is$ ye^{\large pdx}=\int Q e^{\large pdx} dx+c$
Step 1:
$\large\frac{dy}{dx}+\frac{x}{y}$$=\sin (x^2)$
$P(x)=\large\frac{1}{x}$$, Q(x)= \sin (x^2)$
$e^{\large\int pdx}= e^{\int \large\frac{1}{x}}$$ dx$
$\qquad= e^{\large\log x}$
$\qquad=x$
The solution is $yx= \int \sin (x^2) .x dx+c$
$yx=\large\frac{-1}{2} $$\cos x^2+c$

 

answered Sep 6, 2013 by meena.p
edited Sep 6, 2013 by meena.p
 
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