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The solution of differential equation (xdy/dx)=y+x2 is
- a)y=logx+(x2/2)+a
- b)y=(x2/3)+(a/x)
- c)y=x2+ax
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The solution of differential equation (xdy/dx)=y+x2 isa)y=logx+(x2/2)+...
Solution:
Given differential equation is (xdy/dx)=y x2
Separating variables on both sides, we get ydy/x = x dx
Integrating both sides, we get ∫ydy = ∫x dx2
Solving the above integrals, we get (y2/2) = (x3/3) + c, where c is the constant of integration.
Therefore, the solution of the given differential equation is y = ±√(2(x3/3 + c))
But as the given options do not match with the above solution, we need to simplify the above solution further.
Putting the constant of integration c = a/2, we get
y = ±√(2(x3/3 + a/2))
=> y = ±√(2/3) (x3 + 3a)/2
Taking the positive sign, we get y = √(2/3) (x3 + 3a)/2
Taking a common from the square root, we get y = √(2/3)(1/2) (x3 + 3a)
=> y = √(1/3) (x3 + 3a)
On simplifying the above equation, we get
y = (x2/2) √(4a + x6)/x3
Hence, the correct option is (a) y = (x2/2)logx (x2/2)
Given differential equation is (xdy/dx)=y x2
Separating variables on both sides, we get ydy/x = x dx
Integrating both sides, we get ∫ydy = ∫x dx2
Solving the above integrals, we get (y2/2) = (x3/3) + c, where c is the constant of integration.
Therefore, the solution of the given differential equation is y = ±√(2(x3/3 + c))
But as the given options do not match with the above solution, we need to simplify the above solution further.
Putting the constant of integration c = a/2, we get
y = ±√(2(x3/3 + a/2))
=> y = ±√(2/3) (x3 + 3a)/2
Taking the positive sign, we get y = √(2/3) (x3 + 3a)/2
Taking a common from the square root, we get y = √(2/3)(1/2) (x3 + 3a)
=> y = √(1/3) (x3 + 3a)
On simplifying the above equation, we get
y = (x2/2) √(4a + x6)/x3
Hence, the correct option is (a) y = (x2/2)logx (x2/2)
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The solution of differential equation (xdy/dx)=y+x2 isa)y=logx+(x2/2)+...
Divide by x both side then it will become standard form of linear differential equation then solve that easily
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The solution of differential equation (xdy/dx)=y+x2 isa)y=logx+(x2/2)+ab)y=(x2/3)+(a/x)c)y=x2+axd)none of theseCorrect answer is option 'C'. Can you explain this answer?
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