Question

d^2y/dx^2+3(dy/dx)+ 2y=1+3x+x^2

Answer 1

Answer on Question #77428 – Math – Differential Equations

Question

$$\frac {d ^ {2} y}{d x ^ {2}} + 3 \frac {d y}{d x} + 2 y = 1 + 3 x + x ^ {2}.$$

Solution

The corresponding homogeneous equation is $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$ . So, characteristic equation will be

$$r ^ {2} + 3 r + 2 = (r + 1) (r + 2) = 0.$$

Then the complementary solution will be

$$y _ {c} = C _ {1} e ^ {- x} + C _ {2} e ^ {- 2 x}.$$

The nonhomogeneous equation has $f(x) = 1 + 3x + x^2$ . We will search a particular solution in the next general quadratic polynomial form:

$$y _ {p} = A + B x + C x ^ {2}.$$

Then $\frac{dy}{dx} = B + 2Cx$ and $\frac{d^2y}{dx^2} = 2C$ . Substitute them into the equation:

$$2 C + 3 (B + 2 C x) + 2 (A + B x + C x ^ {2}) = 1 + 3 x + x ^ {2}.$$

The corresponding terms on both sides should have the same coefficients. Hence, we obtain:

$$2 C = 1,$$ $$6 C + 2 B = 3,$$ $$2 C + 3 B + 2 A = 1.$$

Now we have the next solution for unknown coefficients:

$$C = \frac {1}{2},$$ $$B = 0,$$ $$A = 0.$$

The general solution of equation is

$$y = y _ {c} + y _ {p} = C _ {1} e ^ {- x} + C _ {2} e ^ {- 2 x} + \frac {x ^ {2}}{2}.$$

Answer: $y = C_1 e^{-x} + C_2 e^{-2x} + \frac{x^2}{2}$ .

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