Chapter 12: Q. 65 (page 945)

Solve the exact differential equations in Exercises 63–66. $e^{x} \ln y + x^{3} + \frac{e^{x}}{y} \frac{d y}{d x} = 0$

Short Answer

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The solution of given exact differential equation is: $e^{x} \ln y + \frac{x^{4}}{4} + C = 0$

Step by step solution

Step 1. Given information

Exact differential equation, $e^{x} \ln y + x^{3} + \frac{e^{x}}{y} \frac{d y}{d x} = 0$

Step 2. Solving the given exact differential equation

$Given, e^{x} \ln y + x^{3} + \frac{e^{x}}{y} \frac{d y}{d x} = 0 \Rightarrow (e^{x} \ln y + x^{3}) d x + \frac{e^{x}}{y} d y = 0 It is a exact differential equation of the the form M d x + N d y = 0, with \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} . the solution is given by, f (x, y) = \int (treating y a s constant in M) d x + \int (terms independent of x in N) d y = 0 \Rightarrow \int (e^{x} \ln y + x^{3}) d x + \int (0) d y = 0 \Rightarrow e^{x} \ln y + \frac{x^{4}}{4} + C = 0$

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Solve the exact differential equations in Exercises 63–66. $e^{x} \ln y + x^{3} + \frac{e^{x}}{y} \frac{d y}{d x} = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solving the given exact differential equation

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Solve the exact differential equations in Exercises 63–66.exlny+x3+exydydx=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solving the given exact differential equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Solve the exact differential equations in Exercises 63–66. $e^{x} \ln y + x^{3} + \frac{e^{x}}{y} \frac{d y}{d x} = 0$