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Solve the given initial-value problem. The DE is homogeneous. xy2 dx = y3 - x³, y(1) = 1 dx

Solve the given initial-value problem. The DE is homogeneous. xy2 dx = y3 - x³, y(1) = 1 dx

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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PLEASE ANSWER THE QUESTION BADLY NEED TY. THIS IS MATH 156 DIFFERENTIAL EQUATIONS.

Solve the given initial-value problem. The DE is homogeneous. xy² dy y³. ³ - x³, y(1) = 1 dx Step 1 We are given a differential equation and will rewrite it in the form M(x, y) dx + N(x, y) dy = 0. M(x, y) = = = Find the functions M and N. N(x, y): = xy2 dy dx = 3 y³ – xy² dy - (y³x³) dx 0 xy² dy + (-y³ + x³) dx = 0 xy² dy = (³x³) dx = x3
Transcribed Image Text:Solve the given initial-value problem. The DE is homogeneous. xy² dy y³. ³ - x³, y(1) = 1 dx Step 1 We are given a differential equation and will rewrite it in the form M(x, y) dx + N(x, y) dy = 0. M(x, y) = = = Find the functions M and N. N(x, y): = xy2 dy dx = 3 y³ – xy² dy - (y³x³) dx 0 xy² dy + (-y³ + x³) dx = 0 xy² dy = (³x³) dx = x3
If the right side of the equation = f(x, y) can be expressed as a function of the ratio only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. The following method outline can be used for any homogeneous equation. That is, the substitution y = xv(x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by gives the solution to the original equation. (x² + 3xy + y²) dx - x² dy = 0 (a) Show that the given equation is homogeneous. Dividing by x², we see that the equation becomes (1+ (b) Solve the differential equation. (c) Draw a direction field and some integral curves. Are they symmetric with respect to the origin? O Yes O No Jdx dy. Hence the differential equation -Select- homogeneous.
Transcribed Image Text:If the right side of the equation = f(x, y) can be expressed as a function of the ratio only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. The following method outline can be used for any homogeneous equation. That is, the substitution y = xv(x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by gives the solution to the original equation. (x² + 3xy + y²) dx - x² dy = 0 (a) Show that the given equation is homogeneous. Dividing by x², we see that the equation becomes (1+ (b) Solve the differential equation. (c) Draw a direction field and some integral curves. Are they symmetric with respect to the origin? O Yes O No Jdx dy. Hence the differential equation -Select- homogeneous.
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