00:01
In this question we have to find out the general solution of the differential equation y' ' -3y ' -4y is equal to 3 e raised to the power 2t minus 8 e raised to the power t cos 2t.
00:18
Now this is non -homogeneous differential equation and the solution of the non -homogeneous differential equation is of the form yc plus yp where c is the complementary solution and yp is the particular solution.
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First of all let us find out the value of yc.
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So for that we will consider the homogeneous part of the differential equation.
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This will be y' ' -3y ' -4y is equal to 0.
00:51
So auxiliary equation will be equal to m square minus 3m minus 4 is equal to 0.
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Splitting middle term this will be equal to minus 4m plus m minus 4 is equal to 0.
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Taking m common m minus 4 taking 1 common we are left with m minus 4.
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That means 2 roots are m plus 1 and m minus 4 equal to 0.
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So value of m will be minus 1 comma 4.
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So these are real and distinct roots.
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So y of t this is the complementary solution will be of the form c1 e raised to the power m1t m1 is minus 1 so minus t plus c2 e raised to the power m2t m2 is 4 so 4t.
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So this is the value of yc.
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Now let us find out the value of yp.
01:50
So for that we have to use the method of undetermined coefficients.
01:57
If we see the right hand side of the differential equation.
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So we have 3 e raised to the power 2t minus 8 e raised to the power t cos 2t.
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So that means we have 1 e raised to the power 2t term.
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So we will write a e raised to the power 2t.
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Then we have e raised to the power t into cos 2t.
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So for e raised to the power t we will write e raised to the power t.
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For cos 2t we will write cos 2t plus sin 2t.
02:28
And here we will have constant b and c.
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Now if we find out the value of yp dash.
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So this will be 2a e raised to the power 2t plus here we will have to apply product rule.
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So derivative of e raised to the power is e raised to the power t.
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Writing this as common as it is.
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Plus now e raised to the power t as it is...