Chapter 2: Q 2.6-41E (page 77)

Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

Short Answer

Expert verified

${(x - y + 2)}^{2} = C e^{2 x} + 1$

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

The given substitution,

$v = x - y + 2$

Simplify the above equation,

$\begin{array}{l} 1 - v = 1 - (x - y + 2) \\ 1 - v = y - x - 1 \end{array}$

And

role="math" localid="1663933729097" $\begin{matrix} 1 - v = y - x - 1 \\ y = 1 - v + x + 1 \\ y = x - v + 2 \\ \frac{dy}{dx} = 1 - \frac{dv}{dx} \end{matrix}$

Given equation,

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

Substitute $v = x - y + 2, 1 - v = y - x - 1$ and $\frac{dy}{dx} = 1 - \frac{dv}{dx}$ in the above equation,

$1 - \frac{dv}{dx} = 1 - v + {(v)}^{- 1}$

Simplify the above equation,

$\begin{array}{l} - \frac{dv}{dx} = - v + {(v)}^{- 1} \\ \frac{dv}{dx} = v - \frac{1}{v} \\ \frac{dv}{dx} = \frac{v^{2} - 1}{v} \end{array}$

Cross multiplication on both sides in the above equation,

$\begin{array}{l} \frac{v}{v^{2} - 1} dv = dx \\ \frac{2 v}{v^{2} - 1} dv = 2 dx \end{array}$

Step 2: Integrating

Taking integration both sides in the above equation

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

Solve the above equation,

$\begin{matrix} \int \frac{2 v}{v^{2} - 1} dv = 2 \int dx \\ \log (v^{2} - 1) = 2 x + \log C \\ \frac{v^{2} - 1}{C} = e^{2 x} \\ v^{2} - 1 = C e^{2 x} \\ v^{2} = C e^{2 x} + 1 \end{matrix}$

Substitute the value of $v = x - y + 2$ in the above equation,

${(x - y + 2)}^{2} = C e^{2 x} + 1$

Hence the solution is ${(x - y + 2)}^{2} = C e^{2 x} + 1$

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Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

Short Answer

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

Step 2: Integrating

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Question: Use the substitution v=x-y+2 to solve equation (8).dydx=y-x-1+x-y+2-1

Short Answer

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

Step 2: Integrating

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$