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The second order linear differential equations t²y" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = t". a) Verify that y(t) = t" is a solution of t²y" + bty' + cy = 0 if r is a solution of ²+(b-1)r+c=0. b) If r²+ (b- 1)r + c = 0 has two distinct real roots then there are two fundamental solutions. In this case, write then and the general solution. c) If r² + (b − 1)r + c = 0 has two complex roots. Use the relations t" = erint and ea+iß = ea(cos ß+isin 3) to obtain two real solutions (fundamental set) and the general solution. d) If r² + (b-1)r + c = 0 has one real repeated root r₁ then y₁ (t) = t" is a solution. Apply reduction of order method to obtain another solution 2(t) that will constitute

The second order linear differential equations t²y" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = t". a) Verify that y(t) = t" is a solution of t²y" + bty' + cy = 0 if r is a solution of ²+(b-1)r+c=0. b) If r²+ (b- 1)r + c = 0 has two distinct real roots then there are two fundamental solutions. In this case, write then and the general solution. c) If r² + (b − 1)r + c = 0 has two complex roots. Use the relations t" = erint and ea+iß = ea(cos ß+isin 3) to obtain two real solutions (fundamental set) and the general solution. d) If r² + (b-1)r + c = 0 has one real repeated root r₁ then y₁ (t) = t" is a solution. Apply reduction of order method to obtain another solution 2(t) that will constitute

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The second order linear differential equations ty" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = tr. a) Verify that y(t) = t" is a solution of t²y" + bty' + cy = 0 if r is a solution of ² + (b-1)r + c = 0. b) If r² + (b − 1)r + c = 0 has two distinct real roots then there are two fundamental solutions. In this case, write then and the general solution. c) If r² + (b − 1)r + c = 0 has two complex roots. Use the relations t" = ent and ea+iß = eº (cos ß+i sin 3) to obtain two real solutions (fundamental set) and the general solution. d) If r² + (b − 1)r + c = 0 has one real repeated root r₁ then y₁(t) = t¹ is a solution. Apply reduction of order method to obtain another solution y2(t) that will constitute a fundamental set of solutions. Then write the general solution.
Transcribed Image Text:The second order linear differential equations ty" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = tr. a) Verify that y(t) = t" is a solution of t²y" + bty' + cy = 0 if r is a solution of ² + (b-1)r + c = 0. b) If r² + (b − 1)r + c = 0 has two distinct real roots then there are two fundamental solutions. In this case, write then and the general solution. c) If r² + (b − 1)r + c = 0 has two complex roots. Use the relations t" = ent and ea+iß = eº (cos ß+i sin 3) to obtain two real solutions (fundamental set) and the general solution. d) If r² + (b − 1)r + c = 0 has one real repeated root r₁ then y₁(t) = t¹ is a solution. Apply reduction of order method to obtain another solution y2(t) that will constitute a fundamental set of solutions. Then write the general solution.
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