{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

wksht02sol

# wksht02sol - MATH 152 L1 L2 Applied Linear Algebra and...

This preview shows pages 1–3. Sign up to view the full content.

MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 02 Worksheet Solutions : First-order differential equations Q1. (Change of variables) Consider the first-order differential equation 1 + x y - sin y dy dx = 0 , which is neither linear nor separable. The purpose of this problem is to show you a trick. (a) Instead of thinking of the variable y as a function of x , suppose we think of the variable x as a function of y . The derivative of x with respect to y is dx dy , the reciprocal of dy dx . Rewrite the given equation so that it is an equation for dx dy . Solution x y - sin y = - dx dy , dx dy + x y = sin y. (1) (b) Is your rewritten equation linear, separable or both? Solution The equation (1) is a first-order linear ordinary differential equation for x ( y ). However, it is a non-separable equation. (c) Solve your differential equation for x as a function of y . This gives you an implicit function for y in terms of x . Solution Consider the equation (1) for x ( y ). The integrating factor is exp •Z 1 y dy = e ln y = y. Then consider d dy ( yx ) = y dx dy + x y = y sin y. Integrating with respect to y gives yx = Z y sin y dy = - Z y d (cos y ) = - y cos y + Z cos y dy = - y cos y + sin y + C. The implicit solution for y in terms of x is given by xy + y cos y - sin y = C .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Q2. (Homogeneous equation) In the previous problem, we showed how one sometimes converts nonlinear differential equations into linear differential equations. This problem shows you another trick to convert a certain type of differential equation into a separable equation. Let us consider the equation
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

wksht02sol - MATH 152 L1 L2 Applied Linear Algebra and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online