Question

Find the particular solution of dy/dx+2y=2x^2+3

Answer 1

Answer to Question #88759 – Math – Differential Equations

Question

Find the particular solution of dy/dx+2y=2x^2+3

Solution

$$\frac{dy}{dx} + 2y = 2x^2 + 3 \dots (\mathrm{i})$$

On comparing with,

$$\frac{dy}{dx} + Py = Q$$

We get,

$$P = 2, Q = 2x^2 + 3$$

Then, Integrating factor $(IF) = e^{\int P dx} = e^{\int 2dx} = e^{2x}$

Now multiplying both sides of (i) with $IF$, we get

$$y.IF = \int (2x^2 + 3)IF \, dx$$ $$ye^{2x} = \int (2x^2 + 3)e^{2x} \, dx$$

Apply integrating by parts to right side,

$$ye^{2x} = (2x^2 + 3)\int e^{2x} \, dx - \int \left\{\frac{d}{dx} (2x^2 + 3)\int e^{2x} \, dx\right\} dx$$ $$ye^{2x} = (2x^2 + 3)\frac{e^{2x}}{2} - \int \left\{4x \right\}_{2}^{\sigma^{2x}}$$ $$ye^{2x} = (2x^2 + 3)\frac{e^{2x}}{2} - \int \left\{2xe^{2x}\right\} dx$$

Again Apply integrating by parts,

$$ye^{2x} = (2x^2 + 3)\frac{e^{2x}}{2} - \left[ 2x \int e^{2x} \, dx - \int 2\right]_{2}^{\sigma^{2x}}$$ $$ye^{2x} = (2x^2 + 3)\frac{e^{2x}}{2} - \left[ 2x \right]_{2}^{\sigma^{2x}} \quad \begin{array}{c} \sigma^{2x} \\ 2 \end{array} \quad ] + C \quad \text{[where } C \text{ is constant of integration]}$$ $$ye^{2x} = (2x^2 + 3)\frac{e^{2x}}{2} - xe^{2x} + \frac{e^{2x}}{2} + C$$ $$y = \frac{2x^2 + 3}{2} - x + \frac{1}{2} + Ce^{-2x}.$$

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