Problems on Differential Equation
1. Find the order and degree of \({{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{3}} \right)}^{\frac{6}{5}}}=6y\)
Solution: Given that \({{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{3}} \right)}^{\frac{6}{5}}}=6y\)
\({{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{3}} \right)}^{\frac{6}{5}}}^{\times \frac{5}{6}}={{\left( 6y \right)}^{\frac{5}{6}}}\)
\(\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{3}} \right)={{\left( 6y \right)}^{\frac{5}{6}}}\)
That is \(\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{3}} \right)={{\left( 6y \right)}^{\frac{5}{6}}}\)
The highest differentiation is order that power is degree, hence Order is 2, degree = 1
2. Find the order of the family of the differential equation obtained by eliminating the arbitrary constant b and c from xy = cex + be-x + x²
Solution: Equation of the curve is xy = cex + be-x + x²
Number of arbitrary constant in the given curve is 2
Therefore, the order of the corresponding differential equation is 2
3. Find the order of differential equation of the family of all circle with their centers at the origin
Solution: Given family of the curve x² + y² = a² … (1)
Where a is parameter
From equation differentiation with respect to the x
2x + 2yy₁ = 0
Hence differential equation is x + yy₁ = 0
Order of the differential equation is 1
4. From the differential equation from the relation xy = ax² + b/x ny eliminating the arbitrary constant a, b.
Solution: Given that xy = ax² + b/x … 1
Take LCM
yx² = ax³ + b
Differentiation with respect to the x
x²y₁ + 2xy = 3ax²
xy₁ + 2y = 3ax … 2
Again differentiation with respect to the x
xy₂ + y₁ + 2y₁ = 3ax
xy₂ + 3y₁ = 3a
From equation (2)
xy₁ + 2y = x (xy2 + 3y₁)
xy₁ + 2y = x²y₂ + 3xy₁
x²y₂ + 2xy₁ – 2y = 0
Differential equation x²y₂ + 2xy₁ – 2y = 0