Chapter 9, Problem 1CRE

To determine

To find whether the given equations are linear first order differential equations or not.

Answer to Problem 1CRE

Option (b) is of first order linear differential equation.

Explanation of Solution

Given information:

Given equations are

$\begin{array}{l}(a)y\text{'}=y^5-3x^{4}y\\ (b)y\text{'}=x^5-3x^{4}y\\ (c)y=y\text{'}-3x\sqrt{y}\\ (d)\sin xy\text{'}=y-1\end{array}$

Formula used:

A differential equation of the form $y\text{'}+P(x)y=Q(x)$ where $P(x)$ and $Q(x)$ are continuous function of $x$ is said to be linear differential equation of first order.

1. Equation $y\text{'}=y^5-3x^{4}y$ is of first order but not linear as y has power 5.
2. Equation $y\text{'}=x^5-3x^{4}y$ can be written as $y\text{'}+3x^{4}y=x^5$ is a linear first order differential equation where $P(x)=3x^4$ and $Q(x)=x^5$.
3. Equation $y=y\text{'}-3x\sqrt{y}$ can be written as $y\text{'}-y=3x\sqrt{y}$ is of first order but not linear as $Q(x)=3x\sqrt{y}$ is not a function of $x$ alone.
4. Equation $\sin xy\text{'}=y-1$ can be written as

$\begin{array}{l}y\text{'}=\frac{1}{\sin x}y-\frac{1}{\sin x}\\ y\text{'}-\frac{1}{\sin x}y=\frac{1}{\sin x}\end{array}$

Where $P(x)=-\frac{1}{\sin x}$ and $Q(x)=\frac{1}{\sin x}$ are not continuous on set of real numbers so given equation is not first order linear differential equation.