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Elementary Differential Equations by Dela Fuente, Feliciano, and Uy Chapter 9 Problem 1 — Special Second-Ordered Differential Equations

Find the general solution of the differential equation

ddx(dydx)=6x+3\frac{d}{dx}\left(\frac{dy}{dx}\right)=6x+3

Solution:

ddx(dydx)=6x+3 let u=dydx dudx=6x+3 Integrate,∫dudx=∫(6x+3)dx ∫dudx=6∫xdx+3∫dx u=6x22+3x+C1 u=3x2+3x+C1 Substitute,dydx=3x2+3x+C1 dy=(3x2+3x+C1)dx Integrate,∫dy=∫(3x2+3x+C1)dx ∫dy=3∫x2dx+3∫xdx+C1∫dx y=3x33+3x22+C1x+C2 Simplify,y=x3+3x22+C1x+C2\begin{align*} \frac{d}{dx}\left(\frac{dy}{dx}\right) & =6x+3 \\\ \\ let\:u & =\frac{dy}{dx} \\\ \\ \frac{du}{dx} & =6x+3 \\\ \\ Integrate,\\ \int \frac{du}{dx} & =\int (6x+3)dx \\\ \\ \int \frac{du}{dx} & =6\int xdx+3\int dx \\\ \\ u & =\frac{6x^2}{2}+3x+C_1 \\\ \\ u & =3x^2+3x+C_1 \\\ \\ Substitute, \\ \frac{dy}{dx} & =3x^2+3x+C_1 \\\ \\ dy & =\left(3x^2+3x+C_1\right)dx \\\ \\ Integrate,\\ \int dy & =\int (3x^2+3x+C_1)dx \\\ \\ \int dy & =3\int x^2dx+3\int xdx+C_1\int dx \\\ \\ y & =\frac{3x^3}{3}+\frac{3x^2}{2}+C_1x+C_2 \\\ \\ Simplify, \\ y & =x^3+\frac{3x^2}{2}+C_1x+C_2 \\ \end{align*}
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