The power series solution about x = 0 of the differential equation y\'\' + y = 0

Question

The power series solution about x = 0 of the differential equation y'' + y = 0 is
y = c0 (a0 + a1x + a2x2 + a3x3 + a4x4 + %u2026)
+ c1 (b0 + b1x + b2x2 + b3x3 + a4x4 + %u2026)

Select the matching values for each coefficient. a0

• Drag answer here

a1

• Drag answer here

a2

• Drag answer here

a3

• Drag answer here

a4

• Drag answer here

b0

• Drag answer here

b1

• Drag answer here

b2

• Drag answer here

b3

• Drag answer here

b4

• Drag answer here

Answers to drag are as follows:

• zero
• -1/6
• 1
• -1/2
• 1/24

The power series solution about x = 0 of the differential equation y'' + y = 0 is
y = c0 (a0 + a1x + a2x2 + a3x3 + a4x4 + %u2026)
+ c1 (b0 + b1x + b2x2 + b3x3 + a4x4 + %u2026)

Select the matching values for each coefficient. a0

• Drag answer here

a1

• Drag answer here

a2

• Drag answer here

a3

• Drag answer here

a4

• Drag answer here

b0

• Drag answer here

b1

• Drag answer here

b2

• Drag answer here

b3

• Drag answer here

b4

• Drag answer here

Answers to drag are as follows:

• zero
• -1/6
• 1
• -1/2
• 1/24

zero -1/6 1 -1/2 1/24 a0

• Drag answer here

a1

• Drag answer here

a2

• Drag answer here

a3

• Drag answer here

a4

• Drag answer here

b0

• Drag answer here

b1

• Drag answer here

b2

• Drag answer here

b3

• Drag answer here

b4

• Drag answer here

Explanation / Answer

let y=e^(m*x) is the solution

then m^2+1=0

solving for m ,we get

m=i or -i

where i=sqrt(-1)

so solution =y(x)=A*cos (x)+B*sin (x)

now power series expansion of xos x=1-(x^2/2!)+(x^4/4!)+....

and of sin x=x-(x^3/3!)+(x^5/5!)-....

comparing,we get the coefficients as

a0=1

a1=0

a2=-1/2

a3=0

a4=1/24

b0=0

b1=1

b2=0

b3=-1/6

b4=0

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