Question

Solve the differential Equation
d^2y/dx^2 +3dy/dx -10y=3x^2

Answer 1

Answer on Question #73772 – Math – Differential Equations

Question

Solve the differential Equation

$$\frac {d ^ {2} y}{d x ^ {2}} + 3 \frac {d y}{d x} - 10 y = 3 x ^ {2}$$

Solution

$$y ^ {\prime \prime} + 3 y ^ {\prime} - 10 y = 3 x ^ {2}$$

is a linear nonhomogeneous differential equation

$$y ^ {\prime \prime} + 3 y ^ {\prime} - 10 y = 0$$

is a linear homogeneous differential equation

$$k ^ {2} + 3 k - 10 = 0$$ $$k _ {1} = - 5, \quad k _ {2} = 2$$

$y^{*} = C_{1}e^{-5x} + C_{2}e^{2x}$ – general solution of a linear homogeneous differential equation

We consider the right-hand side of [1]:

$$f (x) = 3 x ^ {2}$$

$\alpha = 0, \beta = 0, \Delta = \alpha \pm \beta i = 0$ does not coincide with solutions [3] of equation, so we seek the particular solution of the nonhomogeneous differential equation in the form

$$\bar {y} = A x ^ {2} + B x + C$$ $$\bar {y} ^ {\prime} = 2 A x + B$$ $$\bar {y} ^ {\prime \prime} = 2 A$$ $$2 A + 6 A x + 3 B - 10 A x ^ {2} - 10 B x - 10 C = 3 x ^ {2}$$ $$- 10 A x ^ {2} + (6 A - 10 B) x + (2 A + 3 B - 10 C) = 3 x ^ {2}$$ $$\left\{ \begin{array}{l} - 10 A = 3 \\ 6 A - 10 B = 0 \\ 2 A + 3 B - 10 C = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} A = - \frac {3}{10} \\ B = - \frac {9}{50} \\ C = - \frac {57}{500} \end{array} \right.$$ $$\bar {y} = - \frac {3}{10} x ^ {2} - \frac {9}{50} x - \frac {57}{500}$$

the general solution of equation [1]:

$$y = y ^ {*} + \bar {y}$$ $$y = C _ {1} e ^ {- 5 x} + C _ {2} e ^ {2 x} - \frac {3}{10} x ^ {2} - \frac {9}{50} x - \frac {57}{500}$$

Answer: $y = C_{1}e^{-5x} + C_{2}e^{2x} - \frac{3}{10} x^{2} - \frac{9}{50} x - \frac{57}{500}$

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