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Problem. Substitution method is a frequently used technique in mathematics. Through an appropriate substitution, a new (and unsolvable) problem can be turned into an old (and solvable) problem. In calculus, we used this method a lot in integrals. This method can be employed in differential equations, either. The differential equation -x+√x² + y² y describes the shape of a plane curve that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See a previous written assignment. This equation can be solved by means of the substitution u = x² + y² with y = y(x). Solve the equation following the steps below. 1. Express du in terms of x, y and dy. (Hint: Use the chain rule.) dy dx 2. Use the differential equation of y(x) to form a differential equation of u(x).

Problem. Substitution method is a frequently used technique in mathematics. Through an appropriate substitution, a new (and unsolvable) problem can be turned into an old (and solvable) problem. In calculus, we used this method a lot in integrals. This method can be employed in differential equations, either. The differential equation -x+√x² + y² y describes the shape of a plane curve that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See a previous written assignment. This equation can be solved by means of the substitution u = x² + y² with y = y(x). Solve the equation following the steps below. 1. Express du in terms of x, y and dy. (Hint: Use the chain rule.) dy dx 2. Use the differential equation of y(x) to form a differential equation of u(x).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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Problem. Substitution method is a frequently used technique in mathematics. Through an appropriate substitution, a new (and unsolvable) problem can be turned into an old (and solvable) problem. In calculus, we used this method a lot in integrals. This method can be employed in differential equations, either. The differential equation dy dx -x + √x² + y² Y describes the shape of a plane curve that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See a previous written assignment. This equation can be solved by means of the substitution u = x² + y² with y = y(x). Solve the equation following the steps below. 1. Express du in terms of x, y and dy. (Hint: Use the chain rule.) 2. Use the differential equation of y(x) to form a differential equation of u(x).
Transcribed Image Text:Problem. Substitution method is a frequently used technique in mathematics. Through an appropriate substitution, a new (and unsolvable) problem can be turned into an old (and solvable) problem. In calculus, we used this method a lot in integrals. This method can be employed in differential equations, either. The differential equation dy dx -x + √x² + y² Y describes the shape of a plane curve that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See a previous written assignment. This equation can be solved by means of the substitution u = x² + y² with y = y(x). Solve the equation following the steps below. 1. Express du in terms of x, y and dy. (Hint: Use the chain rule.) 2. Use the differential equation of y(x) to form a differential equation of u(x).
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3. Solve the differential equation of u(x), then substitute y back to obtain the solution of the original equation of y(x).
Transcribed Image Text:3. Solve the differential equation of u(x), then substitute y back to obtain the solution of the original equation of y(x).
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