## Problems on Differential Equation

**Problems on Differential Equation:**

**1**. Solve

**Solutions: **Given (D^{2}- 5D 6)y = 0

The auxiliary equation is m^{2 }- 5m 6 = 0

i.e m = 2,3

The general solution is given by

y = Ae^{2x} Be^{3x}

**2**. Solve

**Solution:** Given (D^{2} - 6D 3y) = 0.

The auxilary equation is m^{2} - 6m 13 = 0

i.e.

Hence the solution is y = e^{3x} ( ACos 2x B Sin 2x ).

**3**. Solve (D^{2} 1) = 0 given y(0) = 0, y''(0) = 1.

**Solution: **Given (D^{2} 1) = 0

A.E is m^{2} 1 = 0

M = ±i

Y = A cosx B sinx

Y(x) = A cosx B sinx

Y(0) = A = 0

Y'(0) = B = 1

∴ A = 0, B = 1 ; i.e y = (0) cosx sinx

y = sinx.

^{4. Solve }

**Solution: **Given (D^{2} - 4D 13)y = e^{2x}

The Auxiliary equation is m^{2 }- 4m 13 = 0