Chapter 2: Q2.2-14E (page 46)

$I n p r o b l e m 7 - 16, s o l v e t h e e q u a t i o n . \frac{d x}{d t} - x^{3} = x$

Short Answer

Expert verified

$T h e s o l u t i o n o f t h e g i v e n d i f f e r e n t i a l e q u a t i o n i s \frac{x}{1 + x^{2}} = C e^{t} .$

Step by step solution

Concept of Separable Differential Equation

$A f i r s t - o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n \frac{d y}{d x} = f (x, y) i s r e f e r r e d t o a s s e p a r a b l e i f t h e f u n c t i o n i n t h e r i g h t - h a n d s i d e o f t h e e q u a t i o n i s e x p r e s s e d a s a p r o d u c t o f t w o f u n c t i o n s g (x) t h a t i s a f u n c t i o n o f x a l o n e a n d h (y) t h a t i s a f u n c t i o n o f y a l o n e . A s e p a r a b l e d i f f e r e n t i a l e q u a t i o n c a n b e e x p r e s s e d a s \frac{d y}{d x} = g (x) h (y) . B y s e p a r a t i n g t h e v a r i a b l e s, t h e e q u a t i o n f o l l o w s \frac{d y}{h (y)} = g (x) d x . T h e n, o n d i r e c t i n t e g r a t i o n o f b o t h s i d e s, t h e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n i s d e t e r m i n e d .$ $A f i r s t - o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n \frac{d y}{d x} = f (x, y) i s r e f e r r e d t o a s s e p a r a b l e i f t h e f u n c t i o n i n t h e r i g h t - h a n d s i d e o f t h e e q u a t i o n i s e x p r e s s e d a s a p r o d u c t o f t w o f u n c t i o n s g (x) t h a t i s a f u n c t i o n o f x a l o n e a n d h (y) t h a t i s a f u n c t i o n o f y a l o n e . A s e p a r a b l e d i f f e r e n t i a l e q u a t i o n c a n b e e x p r e s s e d a s \frac{d y}{d x} = g (x) h (y) . B y s e p a r a t i n g t h e v a r i a b l e s, t h e e q u a t i o n f o l l o w s \frac{d y}{h (y)} = g (x) d x . T h e n, o n d i r e c t i n t e g r a t i o n o f b o t h s i d e s, t h e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n i s d e t e r m i n e d .$ uncaught exception: Invalid chunk

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A first-order ordinary differential equation $\frac{d y}{d x} = f (x, y)$ is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions g(x) that is a function of x alone and h(y) that is a function of y alone.

A separable differential equation can be expressed as $\frac{d y}{d x} = g (x) h (y)$ . By separating the variables, the equation follows $\frac{d y}{h (y)} = g (x) d x$ . Then, on direct integration of both sides, the solution of the differential equation is determined.

Solution of the Equation

The given equation is

$\frac{d x}{d t} - x^{3} = x . . . . . . (1)$

After separating the variables, equation (1) can be written as

$\frac{d x}{d t} = x + x^{3} = x (1 + x^{2}) \frac{d x}{x (1 + x^{2})} = d t . . . . . (2)$

Before proceeding for integration, let us decompose the integrand

\frac{1}{x (1 + x^{2})}

into partial fraction decomposition. It follows,

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

Then, equation (2) becomes,

$(\frac{1}{x} - \frac{x}{1 + x^{2}}) d x = d t . . . . . (3)$

Now, integrate both sides of equation (3). It results,

$\int (\frac{1}{x} - \frac{x}{1 + x^{2}}) d x = \int d t \int \frac{d x}{x} - \int \frac{x d x}{1 + x^{2}} = \int d t \int \frac{d x}{x} - \frac{1}{2} \int \frac{2 x d x}{1 + x^{2}} = \int d t \int \frac{d x}{x} - \int \frac{d (1 + x^{2})}{1 + x^{2}} = \int d t \ln x - \ln (1 + x^{2}) = t + \ln C [\ln C = \ln i n t e g r a t i n g C o n s \tan t] \ln (\frac{x}{1 + x^{2}}) - \ln C = t \ln (\frac{x}{C (1 + x^{2})}) = t \frac{x}{1 + x^{2}} = C e^{t}$ Therefore, the solution of the Therefore, the solution of given equation is $\frac{x}{1 + x^{2}} = C e^{t}$ .

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$I n p r o b l e m 7 - 16, s o l v e t h e e q u a t i o n . \frac{d x}{d t} - x^{3} = x$

Short Answer

Step by step solution

Concept of Separable Differential Equation

Solution of the Equation

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Inproblem7-16,solvetheequation.dxdt-x3=x

Short Answer

Step by step solution

Concept of Separable Differential Equation

Solution of the Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Probability and Statistics

Statistics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

$I n p r o b l e m 7 - 16, s o l v e t h e e q u a t i o n . \frac{d x}{d t} - x^{3} = x$