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+ P(x)y = f(x) ds dx We are given the following equation. y' = 2y + x² + 3 This can be written in standard form by subtracting the term in y from both sides of the equation. dy - 2y=x²+3 dx Thus, we have the following coefficient functions from the standard form. P(x) = -2 f(x) = x² + (No Response)

+ P(x)y = f(x) ds dx We are given the following equation. y' = 2y + x² + 3 This can be written in standard form by subtracting the term in y from both sides of the equation. dy - 2y=x²+3 dx Thus, we have the following coefficient functions from the standard form. P(x) = -2 f(x) = x² + (No Response)

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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Recall that the standard dy = dx = + P(x)y = f(x) We are given the following equation. y' = 2y + x² + 3 This can be written in standard form by subtracting the term in y from both sides of the equation. dy dx -2 x² + - 2y = x² + 3 Thus, we have the following coefficient functions from the standard form. P(x) f(x) rear rst-order diferential Ton is as follows. (No Response)
Transcribed Image Text:Recall that the standard dy = dx = + P(x)y = f(x) We are given the following equation. y' = 2y + x² + 3 This can be written in standard form by subtracting the term in y from both sides of the equation. dy dx -2 x² + - 2y = x² + 3 Thus, we have the following coefficient functions from the standard form. P(x) f(x) rear rst-order diferential Ton is as follows. (No Response)
A differential equation is said to be separable if it can be written in the form = g(x)h(y). If this is the case, we can rewrite the equation with all terms and differentials involving x on one side of the equation and those involving y on the other side. dx dy dx = g(x)h(y) dy = g(x)dx h(y) p(y)dy = g(x)dx In the last equation, we substitute p(y) = dy dx Separate the variables for the given differential equation. 4y + 3)2 8x + dy dx = = dy = (4y + 3)² (8x + 5)² 1 (8x + 5)² 1 h(y) dx for convenience.
Transcribed Image Text:A differential equation is said to be separable if it can be written in the form = g(x)h(y). If this is the case, we can rewrite the equation with all terms and differentials involving x on one side of the equation and those involving y on the other side. dx dy dx = g(x)h(y) dy = g(x)dx h(y) p(y)dy = g(x)dx In the last equation, we substitute p(y) = dy dx Separate the variables for the given differential equation. 4y + 3)2 8x + dy dx = = dy = (4y + 3)² (8x + 5)² 1 (8x + 5)² 1 h(y) dx for convenience.
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To find the solution of an equation in standard form, we first find a specific function called the integrating factor. Then, both sides of the equation are multiplied by this function. Finally, we integrate both sides of the equation to determine the solution. We have written the linear equation in standard form and have identified the coefficient functions. dx - 2y = x² + 3, P(x) = -2, f(x) = x² + 3 Next, we must find the integrating factor for this equation, which is defined as follows. μ(x) = e/P(x) dx Evaluating the integral that appears in the exponent results in the following. (Do not add the constant of integration, as it will not be used in defining u(x). This is because both sides of the equation will be multiplied by u(x), so a factor involving the constant may be divided out of each side of the equation.) [P(x) dx = [-2 dx Therefore, the integrating factor is as follows. μ(x) = e/P(x) dx
Transcribed Image Text:To find the solution of an equation in standard form, we first find a specific function called the integrating factor. Then, both sides of the equation are multiplied by this function. Finally, we integrate both sides of the equation to determine the solution. We have written the linear equation in standard form and have identified the coefficient functions. dx - 2y = x² + 3, P(x) = -2, f(x) = x² + 3 Next, we must find the integrating factor for this equation, which is defined as follows. μ(x) = e/P(x) dx Evaluating the integral that appears in the exponent results in the following. (Do not add the constant of integration, as it will not be used in defining u(x). This is because both sides of the equation will be multiplied by u(x), so a factor involving the constant may be divided out of each side of the equation.) [P(x) dx = [-2 dx Therefore, the integrating factor is as follows. μ(x) = e/P(x) dx
A differential equation is said to be separable if it can be written in the form dy = g(x)h(y). If this is the case, we can rewrite the equation with all terms and differentials involving x on one side of the equation and those involving y on the other side. dx dy dx = g(x)h(y) dy = g(x)dx h(y) p(y)dy = g(x) dx In the last equation, we substitute p(y) = (4y+3) ² Separate the variables for the given differential equation. 4y+ 3 8x + X dx dy dx Jay = = = (4y + 3)² (8x + 5)² 1 (8x + h(y) 5)² for convenience. dx
Transcribed Image Text:A differential equation is said to be separable if it can be written in the form dy = g(x)h(y). If this is the case, we can rewrite the equation with all terms and differentials involving x on one side of the equation and those involving y on the other side. dx dy dx = g(x)h(y) dy = g(x)dx h(y) p(y)dy = g(x) dx In the last equation, we substitute p(y) = (4y+3) ² Separate the variables for the given differential equation. 4y+ 3 8x + X dx dy dx Jay = = = (4y + 3)² (8x + 5)² 1 (8x + h(y) 5)² for convenience. dx
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