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Differential Calculus> solve the differential equation (D 3 + 1)...

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solve the differential equation (D³ + 1)y = cos² (x/2) + e^-x

parth , 7 Years ago

Grade 12

1 Answers

Vandna srivastava rathor

general solution =Complementary Function+Particular integral

for Complementary Function m³+1=0

(m+1)(m²+m+1)=0

m=-1, $\frac{-1\pm \sqrt{3}}{2}$

so cf= $c_{1}e^{-x}+e^{-x/2} (c_{2}cos\frac{\sqrt{3}}{2}x+c_{3}sin\frac{\sqrt{3}}{2}x)$

for pi =Particular integral

$(D^{3} + 1)y = cos^2 (x/2) + e^{-x}$

$\frac{1}{D^{3} + 1 }\frac{cos(x) }{2}+\frac{1}{D^{3} + 1 }\frac{1 }{2}e^0x+\frac{1}{D^{3} + 1 }e^{-x}$
$-\frac{1 }{2}\frac{D+1}{(D -1)(D+1) }cos(x)+\frac{1 }{2}+x\frac{1}{3(-1)^{2} }e^{-x}$

$-\frac{1 }{2}\frac{D+1}{(D^2 -1) }cos(x)+\frac{1 }{2}+x\frac{1}{3}e^{-x}$

$-\frac{1 }{2}\frac{D+1}{(-1 -1) }cos(x)+\frac{1 }{2}+x\frac{1}{3}e^{-x}$

$\frac{1 }{4}(D+1)cos(x)+\frac{1 }{2}+x\frac{1}{3}e^{-x}$

$\frac{1}{4}(-sinx+cosx)+\frac{1 }{2}+x\frac{1}{3}e^{-x}$

T.S= cf+pi = $c_{1}e^{-x}+e^{-x/2} (c_{2}cos\frac{\sqrt{3}}{2}x+c_{3}sin\frac{\sqrt{3}}{2}x)+\frac{1}{4}(cosx-sinx)+\frac{1 }{2}+x\frac{1}{3}e^{-x}$

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