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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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \[ \frac{dy}{dx} = \frac{-x + \sqrt{x^2 + y^2}}{y} \] 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 \( u = x^2 + y^2 \) with \( y = y(x) \). Solve the equation following the steps below. 1. Express \(\frac{du}{dx}\) in terms of \(x, y\) and \(\frac{dy}{dx}\). (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 \[ \frac{dy}{dx} = \frac{-x + \sqrt{x^2 + y^2}}{y} \] 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 \( u = x^2 + y^2 \) with \( y = y(x) \). Solve the equation following the steps below. 1. Express \(\frac{du}{dx}\) in terms of \(x, y\) and \(\frac{dy}{dx}\). (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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**Instruction for Solving Differential Equations** 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:**Instruction for Solving Differential Equations** 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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