Definitions [1]
A linear differential equation of first order and first degree is
\[\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{P}y=\mathrm{Q}\], where P and Q are the functions of x or constants. Its general solution is \[y.\left(\mathrm{I.F.}\right)=\int\mathrm{Q.}\left(\mathrm{I.F.}\right)\mathrm{d}x+\mathrm{c}\] and the function \[\mathrm{e}^{\int\mathrm{Pdx}}\] is called the integrating factor (I.F.) of the given equation.
Key Points
(i) Write the equation in the form dy/dx + Py = Q
(ii) Identify P and Q
(iii) Find I.F. = \[\mathrm{e}^{\int\mathrm{Pdx}}\]
(iv) Multiply the whole equation by I.F.
(v) Integrate and get a solution.
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Some equations are not linear in the given variables.
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By a suitable change of variables, they can be reduced to linear equations.
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After substitution, the equations become linear in the new variables.
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Denominators must not be zero.
Important Questions [10]
- Evaluate ∫ ∞ 0 5 − 4 X 2 D X
- Prove that for an Astroid X 2 3 + Y 2 3 = a 2 3 the Line 𝜽=𝝅/𝟔 Divide the Arc in the First Quadrant in a Ratio 1:3.
- Evaluate ∫ 1 0 X 5 Sin − 1 X D X and Find the Value of β ( 9 2 , 1 2 )
- Solve : ( 1 + Log X . Y ) D X + ( 1 + X Y ) Dy=0
- Solve the Ode ( Y + 1 3 Y 3 + 1 2 X 2 ) D X + ( X + X Y 2 ) D Y = 0
- Solve Y D X + X ( 1 − 3 X 2 Y 2 ) D Y = 0
- Solve ( Y − X Y 2 ) D X − ( X + X 2 Y ) D Y = 0
- Solve X 2 D 2 Y D X 2 + 3 X D Y D X + 3 Y = Log X . Cos ( Log X ) X
- Solve X Y ( 1 + X Y 2 ) D Y D X = 1
- Evaluate ∫ 6 0 d x 1 + 3 x by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule.
