Any set y1, y2,...,yn of linearly independent solutions of the homogeneous linea

Question

Any set y1, y2,...,yn of linearly independent solutions of the homogeneous linear nth-order differential equation an(x)dny/dxn + an-1(x)dn-1y/dxn-1 + ... + a1(x)dy/dx + a0(x)y = 0 on an interval I is said to be a fundamental set of solutions on the interval. Let f1, f2,...,fn be a fundamental set of solutions of the homogeneous linear nth-order differential equation aX(X)dny/dXn + a(X) on an interval I. Then the general solution on the interval is y = c1f1(x) + c2f2(x) + ... + cnfn(x) where ci, i = 1,2,...n are arbitrary constants. What the definition and theorem above implies is that given a set of n linearly independent solutions of an nth-order homogeneous differential equation then any linear combination of the solutions will also be a solution. Show that if f1(x) = e alpha x and f2(x) = e-alpha x are two solutions of a homogeneous linear differential equation, then f3(x) = sinh alphax and f4(x) = cosh alpha x are also solutions of the same differential equation. Hint: Show that f3 and f4 are linear combinations of f1 and f2. You might look-up the definition of "sinh x" and "coshx".

Explanation / Answer

we know that : sinx = [e^x - e^(-)x] / 2.............and cosx = [e^x + e^(-)x] / 2............so,.......... sinax = [e^ax - e^(-1)ax] / 2 &............ cosax = [e^ax + e^(-)ax] / 2...........hence I have shown that sinax and cosax is the linear combination of e^ax & e^(-)ax............

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