Chapter 2: Q 3RP (page 79)

Question: In Problems 1-30, solve the equation.
$(x^{2} - 2 y^{- 3}) dy + (2 xy - 3 x^{2}) dx = 0$

Short Answer

Expert verified

The solution of the given equation is $x^{2} y - x^{3} + y^{- 2} = C$ .

Step by step solution

Given information and simplification

Given that, $(x^{2} - 2 y^{- 3}) dy + (2 xy - 3 x^{2}) dx = 0 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Let us check whether the given equation is exact or not.

Then, $M = 2 xy - 3 x^{2}, N = x^{2} - 2 y^{- 3}$ .

Differentiate the value of M and N.

$\begin{array}{l} \frac{\partial M}{\partial y} = 2 x \\ \frac{\partial N}{\partial x} = 2 x \\ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \end{array}$

So, the given equation is exact.

Evaluation method

Now, let us assume $M = \frac{\partial F}{\partial x} = 2 xy - 3 x^{2}$ .

Integrate on both sides.

$\begin{matrix} F = \int (2 xy - 3 x^{2}) dx \\ = x^{2} y - x^{3} + g (y) \end{matrix}$

Differentiate the F with respect to y.

$\begin{matrix} \frac{\partial F}{\partial y} = x^{2} + g' (y) \\ = N \end{matrix}$

Equalise the values of N.

$\begin{matrix} x^{2} + g' (y) = x^{2} - 2 y^{- 3} \\ g' (y) = - 2 y^{- 3} \end{matrix}$

Integrate on both sides.

$\begin{matrix} \int g' (y) = - \int 2 y^{- 3} dy \\ g (y) = y^{- 2} + C_{1} \end{matrix}$

Substitute in equation of F.

$\begin{matrix} x^{2} y - x^{3} + y^{- 2} + C_{1} = 0 \\ x^{2} y - x^{3} + y^{- 2} = C \end{matrix}$

So, the solution is role="math" localid="1664022924236" $x^{2} y - x^{3} + y^{- 2} = C$

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Question: In Problems 1-30, solve the equation.
$(x^{2} - 2 y^{- 3}) dy + (2 xy - 3 x^{2}) dx = 0$

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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Question: In Problems 1-30, solve the equation. x2-2y-3dy+2xy-3x2dx=0

Short Answer

Step by step solution

Given information and simplification

Evaluation method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Statistics

Probability and Statistics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: In Problems 1-30, solve the equation.
$(x^{2} - 2 y^{- 3}) dy + (2 xy - 3 x^{2}) dx = 0$