Question

Solve the following ordinary differential equation:
(a) dy/dx=(y-x)/(x-4y)
(b) (2yx^2+4)dy/dx + (2y^2x-3)=0
(c) y"+3y'-10y=3x^2

Answer 1

Answer on Question # 74943, Math-Differential Equations:

Question: Solve the following ordinary differential equation:

(a) $\frac{dy}{dx} = \frac{y - x}{x - 4y}$

(b) $(2y x^{2} + 4) \frac{dy}{dx} + (2y^{2} x - 3) = 0$

(c) $y'' + 3y' - 10y = 3x^2$

Solution: (a). $\frac{dy}{dx} = \frac{y - x}{x - 4y}$

This equation is exact equation i.e. write in the form $M \, dx + N \, dy = 0$

$$M = y - x \quad \text{and} \quad N = x - 4y$$

So, $\Psi = \int (N \, dy) = xy - 2y^2 + C$ (C is integration constant)

Now replace C with $m(x)$, as x was treated as a constant.

$$\Psi = xy - 2y^2 + m(x)$$

Now compare the value of $\frac{\partial}{\partial x} (xy - 2y^2 + m(x))$ and $(y - x)$

So, $\frac{\partial}{\partial x} (xy - 2y^2 + m(x)) = (y - x)$

$$m(x) = D \text{ (constant)}$$

So, $\Psi = xy - 2y^2 + D$

(b). $(2y x^{2} + 4) \frac{dy}{dx} + (2y^{2} x - 3) = 0$

This equation is in exact form i.e. $M(x, y) + N(x, y) \frac{dy}{dx} = 0$

Here, $M(x, y) = 2y^2 x - 3$ and $N(x, y) = 2y x^2 + 4$

$$\Psi = \int (N \, dy) = 4y + x^2 y^2 + c \quad \text{(c = integration constant)}$$

Now replace c with $m(x)$, as x was treated as a constant.

So, $\Psi = 4y + x^2 y^2 + m(x)$

Now compare the value of $\frac{\partial}{\partial x} (4y + x^2 y^2 + m(x)) = 2x y^2 - 3$

So we get, $m(x) = -3x + c_1$ (c₁ is another constant)

So, $\Psi = 4y + x^2 y^2 - 3x + c_1$

(c). $y'' + 3y' - 10y = 3x^2$

To find complementary solution, we put $y'' + 3y' - 10y = 0$ ...(1)

Solution of equation (1) becomes, $y = Ce^{2x} + D e^{-5x}$

Now particular solution is $z = -3\frac{x^2}{10} - \frac{9x}{50} - \frac{57}{500}$

So, the total solution is $\Psi = y + z = Ce^{2x} + D e^{-5x} - 3\frac{x^2}{10} - \frac{9x}{50} - \frac{57}{500}$

Where C and D are constants.

Answer: So, the answers are (a). $\Psi = xy - 2y^2 + D$ , (b). $\Psi = 4y + x^2y^2 - 3x + c_1$ ,

(c). $\Psi = Ce^{2x} + D e^{-5x} - 3\frac{x^2}{10} - \frac{9x}{50} - \frac{57}{500}$ .

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