**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 = ce^{x} + be^{-x} + x²

**Solution:** Equation of the curve is xy = ce^{x} + 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 (xy_{2} + 3y₁)

xy₁ + 2y = x²y₂ + 3xy₁

x²y₂ + 2xy₁ – 2y = 0

Differential equation x²y₂ + 2xy₁ – 2y = 0