Variable Separable
- Some differential equations can be solved by the method of separation of variables.
- (dy/dx) = f(x, y) is a first order and first degree differential equation. If f(x, y) can be written as h(x).g(y), where h(x) is a function of x and g(y) is a function of y, then we can separate h(x) and g(y).
- Solutions of differential equations of the type (dy/dx) = f(ax + by + c):
Step1: Substitute ax + by + c = v in the given differential equation and change the differential coefficients accordingly.
Step2: Obtain the relation f(v)dv = g(x)dx
Step3: Integrate the obtained equation to get a relation between x, v and arbitrary constants.
Step4: Substitute the value of v in the relation obtained in step-3 to get a relation between x, y and arbitrary constants.
Keywords: Variable Separable method, Solutions of differential equations
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Q1Marks:1
Answer:
Explanation:
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Q2Marks:1
Answer:
Explanation:
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Q3
The general solution of the differential equation
(1 + x2) dy = (1 + y2) dx is
Marks:1Answer:
tan-1y = tan-1x + C.
Explanation:
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Q4
The general solution of the differential equation y(x2y + ex)dx – exdy = 0 is
Marks:1Answer:
x3y + 3ex = Cy
Explanation:
Hence, the solution is
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Q5
The differential equation, (ex + 1)ydy = (y + 1)exdx has the solution
Marks:1Answer:
(1 + y)(ex + 1) = Cey.
Explanation: