Question

Discuss, and illustrate with examples, how to solve differential

equations of the forms $dy/dx=f\left(x\right)$ and $d^{2}y/d{x}^2=f\left(x\right).$

Short answer

The differential equations of the forms,$\frac{dy}{dx}=f\left(x\right)$then$y=\int f\left(x\right)dx+c_1$and$\frac{d^{2}y}{d{x}^2}=f\left(x\right)$, then$y=\int \left(\int e^{x}dx+c_1\right)dx+c_2$.

Step-by-step solution

Definition of differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

Multiplying the equation

Start by multiplying the equation with$dx:$

$$\begin{gathered} \frac{dy}{dx}=f\left(x\right) \\ dy=f\left(x\right)dx \end{gathered}$$

Taking integral on both sides,

$$\begin{gathered} \int dy=\int f\left(x\right)dx \\ y+c_1=\int f\left(x\right)dx \\ y=\int f\left(x\right)dx+c_1 \end{gathered}$$

For example, let$\frac{dy}{dx}=e^x$, then we have:

$$\begin{gathered} y=\int e^{x}dx+c_1 \\ y=e^x+c_1 \end{gathered}$$

Step 3:Second order differentiation.

Further , we have:

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=f\left(x\right) \\ \frac{d}{dx}\left(\frac{dy}{dx}\right)=f\left(x\right) \end{gathered}$$$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=f\left(x\right) \\ \frac{d}{dx}\left(\frac{dy}{dx}\right)=f\left(x\right) \end{gathered}$$

So,

$$\begin{gathered} \frac{dy}{dx}=\int f\left(x\right)dx+c_1 \\ dy=\left(\int f\left(x\right)dx+c_1\right)dx \end{gathered}$$

Taking integral on both sides,

$y=\int \left(\int f\left(x\right)dx+c_1\right)dx+c_2$

For example: if,

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=f\left(x\right) \\ f\left(x\right)=e^x \end{gathered}$$

Then,

$$\begin{gathered} y=\int \left(\int e^{x}dx+c_1\right)dx+c_2 \\ y=\int (e^x+c_1)dx+c_2 \\ y=e^x+c_{1}x+c_2 \end{gathered}$$

Hence,

If, $\frac{dy}{dx}=f\left(x\right)$then $y=\int f\left(x\right)dx+c_1$and If $\frac{d^{2}y}{d{x}^2}=f\left(x\right)$, then $y=\int \left(\int e^{x}dx+c_1\right)dx+c_2$.