Question

A) Find a particular solution to y" + 2y = e^3 + x^3. b) Find the general solution.

Answer 1

Answer:

a.P.I=$\frac{e^{3x}}{11}+\frac{1}{2}(x^3-3x)$

b.G.S=$C_1Cos \sqrt2 x+C_2 Sin\sqrt2 x+\frac{1}{11}e^{3x}+\frac{1}{2}(x^3-3x}$

Step-by-step explanation:

We are given that a linear differential equation

$y''+2y=e^{3x}+x^3$

We have to find the particular solution

P.I=$\frac{e^{3x}}{D^2+2}+\frac{x^3}{D^2+2}$

P.I=$\frac{e^{3x}}{3^2+2}+\frac{1}{2} x^3(1+\frac{D^2}{2})^{-2}$

P.I=$\frac{e^{3x}}{11}+\frac{1-2\frac{D^2}{4}+3\frac{D^4}{16}+...}{2}x^3$

P.I=$\frac{e^{3x}}{11}+\frac{x^3-2\frac{\cdot3\cdot 2x}{4}}{2}+0}$ (higher order terms can be neglected

P.I=$\frac{e^{3x}}{11}+\frac{1}{2}(x^3-3x)$

b.Characteristics equation

$D^2+2=0$

$D=\pm\sqrt2 i$

C.F=$C_1cos \sqrt2x+C_2 sin\sqrt2 x$

G.S=C.F+P.I

G.S=$C_1Cos \sqrt2 x+C_2 Sin\sqrt2 x+\frac{1}{11}e^{3x}+\frac{1}{2}(x^3-3x)$