Monique Slaughter

Answered

2022-01-20

Help!

a) Solve the first order linear differential equation$\frac{dy}{dx}+xy=x,\text{}{y}_{0}=-6$ . b) Write down any two applications of first order linear differential equation.

a) Solve the first order linear differential equation

Answer & Explanation

Annie Gonzalez

Expert

2022-01-20Added 41 answers

a) The given differential equations is,

$\frac{dy}{dx}+xy=x$

where,${y}_{0}=-6$

Now,

$\frac{dy}{dx}=x-xy$

$\Rightarrow \text{}\frac{dy}{dx}=-x(y-1)$

$\Rightarrow \text{}\frac{dy}{y-1}=-xdx$

Integrating we have,

$\mathrm{log}|y-1|==\frac{{x}^{2}}{2}+A$ , A in integrating constant.

Given${y}_{0}=-6$

So,

$\mathrm{log}|-6-1|=0+A$

$\Rightarrow \text{}A=\mathrm{log}|-7|$

Therefore, the solution is,

$\mathrm{log}|y-1|+\frac{{x}^{2}}{2}=\mathrm{log}|-7|$

$\Rightarrow \text{}\mathrm{log}\frac{y-1}{-7}=-\frac{{x}^{2}}{2}$

$\Rightarrow \text{}\frac{y-1}{-7}={e}^{-\frac{{x}^{2}}{2}}$

$\Rightarrow \text{}y-1=-7{e}^{-\frac{{x}^{2}}{2}}$

The rewquired solution is,

$y=1-7{e}^{-\frac{{x}^{2}}{2}}$

where,

Now,

Integrating we have,

Given

So,

Therefore, the solution is,

The rewquired solution is,

einfachmoipf

Expert

2022-01-21Added 32 answers

b) Here we need to write two applications of first order linear differential equation.

So,two applications of first order linear differential equation is,

(ii) Newtons

So,two applications of first order linear differential equation is,

(ii) Newtons

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