Daniell Phillips

Answered

2021-12-26

First-Order Linear Differential Equations; Solutions Suggested by the Equation

$\frac{dy}{dx}={(x+y+1)}^{2}$

Answer & Explanation

Hector Roberts

Expert

2021-12-27Added 31 answers

Step 1

We have given the differential equation as$\frac{dy}{dx}={(x+y+1)}^{2}$ . To solve this differential equation, we have to use substitution to convert the equation into the separable differential equation because separable equations are easy to solve.

Step 2

So substitute$x+y+1=v\text{}\in \text{}\frac{dy}{dx}={(x+y+1)}^{2}$ . First, differentiate $x+y+1=v$ with respect to x to find the expression for $\frac{dy}{dx}$ .

$\frac{d}{dx}(x+y+1)=v$

$1+\frac{dy}{dx}=\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{dv}{dx}-1$

Now, do substitution and simplify the differential equation.

$\frac{dv}{dx}-1={v}^{2}$

$\frac{dv}{dx}=1+{v}^{2}$

$\frac{dv}{1+{v}^{2}}=dx$

Step 3

Now, integrate both sides of$\frac{dv}{1+{v}^{2}}=dx$ and evaluate the integral. Use the formula $\int \frac{du}{1+{u}^{2}}={\mathrm{tan}}^{-1}u+C$ .

$\int \frac{dv}{1+{v}^{2}}=\int dx$

${\mathrm{tan}}^{-1}v=x+C$

${\mathrm{tan}}^{-1}(x+y+1)=x+C$ [Substitute back $x+y+1=v$ ]

Hence, the solution of the given differential equation is${\mathrm{tan}}^{-1}(x+y+1)=x+C$ , where C is an integral constant.

We have given the differential equation as

Step 2

So substitute

Now, do substitution and simplify the differential equation.

Step 3

Now, integrate both sides of

Hence, the solution of the given differential equation is

sirpsta3u

Expert

2021-12-28Added 42 answers

Let.

from eqn(1) and (2)

Vasquez

Expert

2022-01-09Added 457 answers

The given differential equation is

Let

Integrating, we get

Which is the required solution.

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