We have shown that in the case of complex conjugate roots,r =a±ib,b̸=0, of the indicial equation, the general solution to the Cauchy-Euler equation y′′+a1y′+a2y =0, x > 0 is y(x)=xa[c1 cos(b ln x)+c2 sin(b ln x)]. .........................(8.8.26) Q. Show that (8.8.26) can be written in the form y(x)= Axa cos(b ln x −φ) for appropriate constants A and φ.

Q: Determine a suitable form for Y(t) if the method of undetermined coefficients is to be used. y" +…

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Q: Determine a suitable form for Y (t) if the method of undetermined coefficients is to be used. y" +…

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Q: Q.2) The basts of the equatioon y'' + y" = 0,are: b) 1,1.e a) 1,1.e c) 1, x,e²* d) 1,x.e" 0 about

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Q: Find the differential operator that annihilates x cos3x + x e³x sin2x.

A: Given equation is  xcos(3x) +xe3xsin(2x) we have to find the differential operator that annihilates…

Q: x = y + 3x¹ sin² (2) to separable equation then solve it. 1. Reduce the DE. x

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Q: sin(x²)y"-(cosx)y'+x²y = y'-3 y"+5xy'-3y = cosy

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Q: 3. Simplify x(t) = 3 cos(27200t + + sin(27200t + to the form x(t) = a cos(27200t + 0) where a and 0…

A: To convert an expression of the form Acosx ±Bsinxmultiply and divide by A2+B2=A2+B2AcosxA2+B2…

Q: 1. If x'y" + xy' + (x². 4 - -y=0 Y=x sinx , and y= y,v. Which of the following x>0 equations…

A: Solution:    Given:                  x2 y"+ xy' +(x2 14) y =0, x.o                 y1 = x-12…

Q: a) m - 1 b) m – 2 Q.8) The characteristic equation for y"" + 3y" – 4y = 0 , is : b) 2' = 32?-4 c) m…

A: To find out the characteristic equation.

Q: Find the solution of x²y" +5xy' +(4+4x)y=0, x > 0 of the form n =1,2,3,... where co 1. Enter r = =…

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Q: 2. Solve for y in terms of x for the following equations: a) In(1- 2y) = x %3D b) In(y- 1) - In 2 =…

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Q: Find indicial equation at each regular singular point of 2.x (1+x) y" + (3+ x) y' – xy = 0.

A: The given equation can be written as:

Q: Show that each of the functions on the left is a 3-parameter family of solutions of the DE on its…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Solve: (cos x) · (dy/dx) + y sin x = 2x cos2 x, y(Tt/4) = (-15/2-n2)/32.

A: The given differential equation can be written as :            dydx +ytanx = 2xcosx So, the…

Q: Find az/ax and dz/dy. xyz + cos(xy) - 2y sin(x) + z² = 3 Find az/ax and dz/dy. exz ZX y

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Q: Solve Q11

A: 11. Given that the series                      x∑m=1∞mamxm-1+∑m=0∞amxm

Q: a) 12 Q.7) To write the power series £=3 4m as the form -4m a) m - 1 we must shift m to become : d)…

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Q: Q.12) The solution of y" + y' = 0 by using power series method , is! %3! a) y(x) = a, – a,(1-+ b)…

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Q: 3. y" - 2y' = ex (x² + 2) cos2x 4. y" - 2y' - 3y = 4x - 5+ 3xe ²x

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Q: Q.1) The general solution of the equation e' cos , is:

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Q: Solve

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Q: (2 sin(y)) dx + (x y cos(y²)) d y = 0

A: Variable separable differential equation

Q: c) y(x) = c, + cz + Acos3x + Bsin3x Q.6) The singular point (s) of (x² + 4)y" – x y' + 4y = 0 , is…

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Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.