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Show that it is impossible to find continuous functions a₁. differential equation x(t) + a₁(t)x(t) + ao(t)x(t) = 0 has the solution x(t) = t². (Hint: check if the existence and un Consider a function f: R x C → C and the complex initia dz = f(t,z), z(0) = Zo dt or a complex-valued function z: R → C and a complex n

Show that it is impossible to find continuous functions a₁. differential equation x(t) + a₁(t)x(t) + ao(t)x(t) = 0 has the solution x(t) = t². (Hint: check if the existence and un Consider a function f: R x C → C and the complex initia dz = f(t,z), z(0) = Zo dt or a complex-valued function z: R → C and a complex n

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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(a) Show that it is impossible to find continuous functions a₁, ao RR so that the differential equation x(t) + a₁(t)x(t) + ao(t)x(t) = 0 has the solution x(t) = t². (Hint: check if the existence and uniqueness theorem applies.) (b) Consider a function ƒ : R × C → C and the complex initial value problem dz dt = = f(t,z), z(0) and deduce Euler's formula eit dz dt for a complex-valued function z: R → C and a complex number zo. By considering real and imaginary parts of this equation formulate conditions on the function f for (1) to have a unique solution. Show that z(t) = eit and z(t) = cost + i sin t both satisfy Zo <= =iz, z(0) = 1, = cost + i sin t. (1)
Transcribed Image Text:(a) Show that it is impossible to find continuous functions a₁, ao RR so that the differential equation x(t) + a₁(t)x(t) + ao(t)x(t) = 0 has the solution x(t) = t². (Hint: check if the existence and uniqueness theorem applies.) (b) Consider a function ƒ : R × C → C and the complex initial value problem dz dt = = f(t,z), z(0) and deduce Euler's formula eit dz dt for a complex-valued function z: R → C and a complex number zo. By considering real and imaginary parts of this equation formulate conditions on the function f for (1) to have a unique solution. Show that z(t) = eit and z(t) = cost + i sin t both satisfy Zo <= =iz, z(0) = 1, = cost + i sin t. (1)
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