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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.4.19
Chapter 7, Problem 7.4.19

Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²

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1
Rewrite the given differential equation \(y^{2} \frac{dy}{dx} = 3x^{2} y^{3} - 6x^{2}\) to isolate \(\frac{dy}{dx}\). Divide both sides by \(y^{2}\) (assuming \(y \neq 0\)) to get \(\frac{dy}{dx} = 3x^{2} y - \frac{6x^{2}}{y^{2}}\).
Recognize that the equation is separable or can be manipulated into a separable form. Try to express all terms involving \(y\) on one side and all terms involving \(x\) on the other side.
Rewrite the equation as \(\frac{dy}{dx} = 3x^{2} y - 6x^{2} y^{-2}\). This suggests grouping terms to separate variables: \(\frac{dy}{dx} = x^{2} (3y - 6 y^{-2})\).
Express the differential equation in differential form: \(dy = x^{2} (3y - 6 y^{-2}) dx\). Then rearrange to isolate \(y\) terms with \(dy\) and \(x\) terms with \(dx\): \(\frac{dy}{3y - 6 y^{-2}} = x^{2} dx\).
Integrate both sides: integrate \(\int \frac{dy}{3y - 6 y^{-2}}\) with respect to \(y\) and \(\int x^{2} dx\) with respect to \(x\). After integration, include the constant of integration and solve for \(y\) if possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written as a product of a function of y and a function of x, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the solution.
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Algebraic Manipulation of Differential Equations

Before solving, it is often necessary to rearrange and simplify the differential equation by factoring or dividing terms. This step helps isolate dy/dx and express the equation in a form suitable for applying solution methods like separation of variables.
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Integration Techniques

Solving the separated differential equation requires integrating functions of x and y. Familiarity with basic integration rules and techniques, such as power rule and substitution, is essential to find the implicit or explicit solution.
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Integration by Parts for Definite Integrals
Related Practice
Textbook Question

In Exercises 27–32, find dy/dx.

e^(2x)=sin(x+3y)

Textbook Question

In Exercises 1–4, solve for t.

e^(sqrt(t)) = x^2

Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.


17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

Textbook Question

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt,x > 0

Textbook Question

Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).

Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.


19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)