COMPASSLearning

Talk to us Register Login

CBSE Class 12 · 2005 · Differential EquationsLong Answer

Solve the following differential equation:

x \frac{d y}{d x} - y = x^{2} + y^{2}

Answer: $y + x^{2} + y^{2} = C x^{2}$

Solution

Write

\frac{d y}{d x} = \frac{y + x ^{2} + y ^{2}}{x}

(homogeneous). Put

y = v x

:

v + x \frac{d v}{d x} = v + 1 + v^{2} \Rightarrow \frac{d v}{1 + v ^{2}} = \frac{d x}{x}

Integrate:

lo g v + 1 + v^{2} = lo g ∣ x ∣ + lo g C

So

v + 1 + v^{2} = C x

. With

v = \frac{y}{x}

:

\frac{y + x ^{2} + y ^{2}}{x} = C x \Rightarrow y + x^{2} + y^{2} = C x^{2}

CBSE Class 12 Board · 2005 · Subjective

Practise more: all CBSE board PYQs on Differential Equations →

More board questions on Differential Equations

(a) Find the particular solution of the differential equation
$\frac{d y}{d x} - 2 x y = 3 x^{2} e^{x^{2}}; y (0) = 5$

OR
(b) Solve the following differential equation:
$x^{2} d y + y (x + y) d x = 0$
CBSE 2024View solution →
If $\frac{d}{d x} f (x) = 2 x + \frac{3}{x}$ and $f (1) = 1$ , then $f (x)$ is:
CBSE 2023View solution →
Degree of the differential equation $sin x + cos (\frac{d y}{d x}) = y^{2}$ is:
CBSE 2023View solution →
The integrating factor of the differential equation $(1 - y^{2}) \frac{d x}{d y} + y x = a y, (- 1 < y < 1)$ is:
CBSE 2023View solution →
Find the general solution of the differential equation: $(x y - x^{2}) d y = y^{2} d x$
CBSE 2023View solution →
Find the general solution of the differential equation: $(x^{2} + 1) \frac{d y}{d x} + 2 x y = x^{2} + 4$
CBSE 2023View solution →