Podcast
Questions and Answers
The solution of
${dy \over dx} - x tan ( y - x) =$
is __________
The solution of
${dy \over dx} - x tan ( y - x) =$
is __________
- ${sec^2 (y - x) = ce^{x^2 \over 2}}$
- ln $sec^2 (y - x) = {x^2 \over 2} + c$
- ${cos (y - x) = c. e^{x^2 \over 2}}$
- ${sin (y - x) = c. e^{x^2 \over 2}}$ (correct)
An integrating factor of the D. E. :
${(1 + x^3){dy \over dx}+ 3x^2y = x^2}$ is _____
An integrating factor of the D. E. :
${(1 + x^3){dy \over dx}+ 3x^2y = x^2}$ is _____
- ${1 + x^3}$ (correct)
- ${ln (1 + x^3)}$
- ${3ln(1 + x^3)}$
- ${(1 + x^3)^3}$
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Study Notes
Differential Equation
- The provided equation is a first-order differential equation.
- It involves the derivative of a function y with respect to x, denoted by dy/dx.
- The equation also includes trigonometric functions (tan) and algebraic terms (x).
Solution
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The solution to this differential equation represents a family of functions y(x) that satisfy the given equation.
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It is a particular solution and represents the relationship between the variables x and y in terms of an equation.
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Obtaining the solution involves using techniques of solving differential equations, such as:
- Separation of variables
- Integrating factors
- Substitution methods
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Determining the solution would require applying one or more of these methods to the specific equation given.
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The exact solution involves performing integration and potentially solving for y explicitly in terms of x.
Example Solution
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The provided equation is a first-order differential equation.
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It involves the derivative of a function ‘y’‘y’‘y’ with respect to ‘x’‘x’‘x’, denoted by dy/dx{dy/dx}dy/dx.
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The equation also includes trigonometric functions (tan) and algebraic terms (‘x’‘x’‘x’).
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Solving this equation requires using techniques like substitution.
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We begin by substituting u=y−x,du/dx=dy/dx−1u=y-x, \quad {du/dx}={dy/dx}-1u=y−x,du/dx=dy/dx−1
- Substitute into the original equation: du/dx−1=xtan(u).du/dx-1=x\tan(u).du/dx−1=xtan(u).
- Separate variables to solve the differential equation: 1xtan(u)+1du=dx\frac{1}{x\tan(u)+1}du=dx x tan(u)+11du=dx.
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The solution requires non-elementary function integration.The resulting differential equation requires solving techniques involving substitution and variable separation.on: ∫1xtan(u)+1du=∫dx\int \frac{1}{x\tan(u)+1}du=\int dx∫xtan(u)+11du=∫dx gives the solution of y(x) in an implicit form.
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